Thursday, 9 January 2014

Do not put your faith in what statistics say until you have carefully considered what they do not say. ~William W. Watt
Just a few of the many quotes about the use, misuse and fallibility of statistics. To know how to use statistics gives you the ability to separate the wheat from the chaff.
Part 1 of 3: Why You Need to Know.

Statistics are used every day. Have you voted for a politician because he claimed his economic policies would lower the unemployment rate and increase the GDP? Have you chosen a surgery that your doctor said would extend your life expectancy by 10 years if successful, but had a 5% risk of serious side effects? Have you chosen to lower your insurance premiums by $30 per month by increasing the deductible from $500 to $1000? These are some of many everyday situations where a good understanding of statistics can serve as a guide to making better decisions.

1
Work on gaining a knowledge of statistics.
- Average - the usual, or what might be considered ordinary - The 'average' family has 2 children (made up statistic).
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2
Learn the terms most often used in statistical analysis.
3
Start applying them to every day life.
- Newspapers
- Media
- News
- Politics
- Sports
4
Learn about statistics in order to understand what others may be telling you and to facilitate your understanding and ability to know what questions to ask.
5
Learn the best way to represent your statistics, if needed. Part 2 of 3: The Use of Statistics

Room Full of Kids

Set of variables

Providing a given set of variables, this article will explain the process.

1
Find some software that will help you manipulate a given set of values.
2
Insert the values that are displayed in the image. How that is done will depend on your software. Chances are, it will have a grid like appearance.
3
Query the program. Basically, you are asking the program to come up with (in this instance), the mean (5.5), the mode (6), and the median (6).
- Note that the software doesn't have to lay the information out in a straight line to come up with the answer.
4
Understand what it indicates. These numbers indicate that if this was a room full of children (24 of them), that the average age (the mean) is 5.5 years old. The mode of six would indicate that there are more six year old children than any other age in the room. The median is indicated by taking the set of (1,1,1,2,3,4,4,5,5,5,6,6,6,6,6,7,7,7,7,8,8,8,9,9) and counting 12 in from each side. In this instance, it puts you straight in the middle of the sixes and therefore the median is six. You would add the numbers (in this case 6+6), then divide by two, which of course, would be 6.

Seasonal Sports Statistics

Sports statistics and the understanding and manipulation thereof, can make or break a person's wallet. Thousands of dollars can be lost on the basis of a point spread. Statistics is the bread and butter of the sports world.

1
Decide what you are going to use the statistics for. You can use stats to tell about percentage of games won, percentage of games won versus another team, etc.
2
Go to your newspaper for the information or your team's web page or sports web page.
3
Calculate the win percentage. Divide the number of wins by the total number of games.
4
Calculate the loss percentage. Divide the number of losses by the total number of games.
5
Example:
- Minnesota Vikings have a current tally of 1 win and 3 losses out of 4 games.
  - Divide one by four to get a win percentage of 25%.
  - Divide three by four to get a loss percentage of 75%.
- Washington Redskins with 3 wins and 2 losses. This would, seemingly, be comparing apples and oranges, because of the different number of games played.
  - Divide three by five for a win percentage of 60%.
  - Divide two by five for a loss percentage of 40%.
- With this information, you can tell that the Redskins are easily outplaying the Vikings at this juncture.

Living Statistics

Living statistics would accomplish stats like Cost of Living, Employment Rate, Crime Rate among many other things.

1
Pick a site with statistical information about various cities.
2
Determine the median income of your county.
- In the 2000 census, Thurston County, Washington had a median income of $46,975, compared to a national median income of $41,994. By dividing 46,975 by 41,994, you can figure out that Thurston County's median income is 11% higher than the national median income.
- Also in the 2000 census, McDowell County, West Virginia's median income was $16,931 and the national median income was $41,994. Divide 16,931 by 41,994 to see that McDowell County is 40% below the national median.

Crime Statistics

Statistics are used in regards to crime rates, increases and decreases of recidivism, and many other ways.

Political Statistics

As in sports, the ability to use statistical analysis in a positive fashion and use them to get your message across are career makers and career enders. A good political statistician can, for all intents and purposes, 'write his own ticket'.

Part 3 of 3: Definitions

Mode - the most frequent occurrence of a variable in a set or sampling of variables.
- The mode of the sample [1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17] is 6.
Median- Relates to, or constitutes the middle value of a distribution.
- Find the Median of: 9, 3, 44, 17, 15 (Odd amount of numbers)
  - Line up your numbers: 3, 9, 15, 17, 44 (smallest to largest)
  - The Median is: 15 (The number in the middle)
- Find the Median of: 8, 3, 44, 17, 12, 6 (Even amount of numbers)
  - Line up your numbers: 3, 6, 8, 12, 17, 44
  - Add the 2 middles numbers and divide by 2: 8+12 = 20 ÷ 2 = 10
  - The Median is 10.
Standard Deviation- A measure of the spread of the values in a given set. The higher the standard deviation, the less numbers in the set tend to cluster near the mean.
Distribution - Statistical data arranged to show the frequency with which the possible values of a variable occur.
Bell-shaped curve- The curve representing a continuous frequency distribution with a shape having the overall curvature of the vertical cross section of a bell; usually applied to the normal distribution.
Probability - The measure of how likely something is going to occur (e.g. the chances of the coin landing on heads is 1/2, the chances of a dice rolling in a certain number is 1/6).
Outliers - are the numbers that can feasibly throw off the statistics because they are 'one offs'. By that, they are atypical of the rest of the data.

Warnings

Statistics are only as good or as bad as the person wielding them. Take every use of statistics with a grain of salt and use your knowledge of statistics to come to a decision of your own.

INTRODUCTION TO STATISTICS ---- By. Mwl. Japhet Masatu.

What Is Statistics?

Everyday we encounter information. Information comes at us in a number of different ways and in different forms. How many calories did each of us eat for breakfast? How far from home did everyone travel today? How big is the place that we call home? How many other people call it home? And that’s only the beginning of the information that is out there. To make sense of all of this information, certain tools and ways of thinking are necessary. The mathematical science called statistics is what helps us to deal with this information overload.
Statistics is the study of numerical information, called data. Statisticians acquire, organize, and analyze data. Each part of this process is also scrutinized. The techniques of statistics are applied to a multitude of other areas of knowledge. Below is an introduction to some of the main topics throughout statistics.

Populations and Samples

One of the recurring themes of statistics is that we are able to say something about a large group based upon the study of a relatively small portion of that group. The group as a whole is known as the population. The portion of the group that we study is the sample.
As an example of this, suppose we wanted to know the average height of people living in the United States. We could try to measure over 300 million people, but this would be infeasible. It would be a logistical nightmare conduct the measurements in such a way that no one was missed and no one was counted twice.
Due to the impossible nature of measuring everyone in the United States, we could instead use statistics. Rather than finding the heights of everyone in the population, we take a statistical sample of a few thousand. If we have sampled the population correctly, then the average height of the sample will be very close to the average height of the population.

Acquiring Data

To draw good conclusions, we need good data to work with. The way that we sample a population to obtain this data should always be scrutinized. Which kind of sample we use depends on what question we’re asking about the population. The most commonly used samples are:

Simple Random
Stratified
Clustered

It’s equally important to know how the measurement of the sample is conducted. To go back to the above example, how do we acquire the heights of those in our sample?

Do we let people report their own height on a questionnaire?
Do several researchers throughout the country measure different people and report their results?
Does a single researcher measure everyone in the sample with the same tape measure?

Each of these ways of obtaining the data has its advantages and drawbacks. Anyone using the data from this study would want to know how it was obtained

Organizing the Data

Sometimes there is a multitude of data, and we can literally get lost in all of the details. It’s hard to see the forest for the trees. That’s why it’s important to keep our data well organized. Careful organization and graphical displays of the data help us to spot patterns and trends, before we actually do any calculations.

Since The way that we graphically present our data depends upon a variety of factors. Common graphs are:

Pie charts or circle graphs
Bar or pareto graphs
Scatterplots
Time plots
Stem and leaf plots
Box and whisker graphs

In addition to these well known graphs, there are others that are used in specialized situations.

Descriptive Statistics

One way to analyze data is called descriptive statistics. Here the goal is to calculate quantities that describe our data. Numbers called the mean, median and mode are all used to indicate the average or center of the data. The range and standard deviation are used to say how spread out the data is. More complicated techniques, such as correlation and regression describe data that is paired.

Inferential Statistics

When we begin with a sample and then try to infer something about the population, we are using inferential statistics. In working with this area of statistics, the topic of hypothesis testing arises. Here we see the scientific nature of the subject of statistics, as we state a hypothesis, then use statistical tools with our sample to determine the likelihood that we need to reject the hypothesis or not. This explanation is really just scratching the surface of this very useful part of statistics.

Applications of Statistics

It is no exaggeration to say that the tools of statistics are used by nearly every field of scientific research. Here are a few areas that rely heavily on statistics:

Psychology
Economics
Medicine
Advertising
Demography

The Foundations of Statistics

Although some think of statistics as a branch of mathematics, it is better to think of it as a discipline that is founded upon mathematics. Specifically, statistics is built up from the field of mathematics known as probability. Probability gives us a way to determine how likely an event is to occur. It also gives us a way to talk about randomness. This is key to statistics because the typical sample needs to be randomly selected from the population.
Probability was first studied in the 1700s by mathematicians such as Pascal and Fermat. The 1700s also marked the beginning of statistics. Statistics continued to grow from its probability roots, and really took off in the 1800s. Today it’s theoretical scope continues to be enlarged in what is known as mathematical statistics.

STATISTICS CALCULATIONS -- By. Mwl. Japhet Masatu.

HOW TO CALCULATE SIMPLE STATISTICS.

Introduction

In statistics, the most important calculations are the mean, mode, median, variance, and standard deviation (std dev)
In this hub, I will cover the following:

sample size
population
mean
mode
median
variance
standard deviation

Sample Size and Population

Statistics begins with a set of numbers which are called the sample.
The set of all possible numbers is called the population.
Let's say that we ask 5 friends to rate a popular movie on the scale from 1 to 10.
Then, the sample size is 5 and the population is the set of all people who have seen or will see the movie.

Calculating the mean

So, we ask our 5 friends to rate and movie and here's what we get:
Fred: 6
Sally: 9
Michael: 8
Raul: 9
Elena: 2
To calculate the mean, you sum up all the numbers in the sample and then divide by the sample size. The sum is 5+10+8+9+2= 34. Since the sample size is 5, the mean is 34/5 = 6.8.
This then is the average of the sample.

Calculating the mode

The mode is the number that appears the most often in the sample.
To calculate the mode, we count the number of times each rating is made. So we have one 6, two 9's, one 8, and one 2. Since we have two 9's and one of everything else, 9 is the mode.
But what would happen if we have the following sequence: 2,2,8,9,9?
In this case, we would say that there is no unique mode. A mode is unique if and only if one number is more frequent than all others.

Calculating the median.

The median is the value we get when we order all of our numbers and then find the one in the middle.
If we order the numbers from smallest to largest, we get: 2, 6, 8, 9, 9
Since we have a sample size of 5, the number in the middle is 8.
But what happens if the sample size is even. In this case, we can add the two middle numbers and divide by 2.
So, if our numbers are: 2,6,8,9, then the median is (6+8)/2 = 7.

Calculating Variance.

The variance is a measure of the variation of the sample data. The larger the variance, the more random the answers appear. Many people find standard deviation to be a more useful measure of variability.
The method for calculating the variance is different depending on whether we are calculating the variance of a population (everyone) or the variance of a sample (some but not all).
Here are the steps:
(1) Figure out the mean. This is the sum of the numbers given divided by the sample size (i.e. the average).
(6+ 9+ 8 + 9 + 2)/5 = 34/5 = 6.8
(2) Figure out the difference between each number and its mean so that we have:
(6 - 6.8), (9 - 6.8), (8 - 6.8), (9 - 6.8), (2 - 6.8) = -0.8, 2.2, 1.2, 2.2, -4.8
(3) Get the square of each difference in step #2 so that we have:
(-0.8)*(-0.8), (2.2)*(2.2), (1.2)*(1.2), (2.2)*(2.2), (-4.8)*(-4.8) = 0.64, 4.84, 1.44, 4.84, 23.04
(4) Get the sum of all the squares in step #3 so that we have:
sum of squares = 0.64 + 4.84 + 1.44 + 4.84 + 23.04 = 34.8
(5) Now, for the sample variance, we divide the sum in step #4 by the sample size - 1
Variance = 34.8/(5-1) = 34.8/4 = 8.7

Calculating the Standard Deviation.

The standard deviation, like variance, is a measure of the variation of the sample data. The larger the standard deviation, the more random the answers appear. Standard deviation is more popular as a measure than variance.
The method for calculating the standard deviation is different depending on whether we are calculating the variance of a population (everyone) or the variance of a sample (some but not all). The method is the same as variance with one additional step.
Here are the steps:
(1) Figure out the mean. This is the sum of the numbers given divided by the sample size (i.e. the average).
(6+ 9+ 8 + 9 + 2)/5 = 34/5 = 6.8
(2) Figure out the difference between each number and its mean so that we have:
(6 - 6.8), (9 - 6.8), (8 - 6.8), (9 - 6.8), (2 - 6.8) = -0.8, 2.2, 1.2, 2.2, -4.8
(3) Get the square of each difference in step #2 so that we have:
(-0.8)*(-0.8), (2.2)*(2.2), (1.2)*(1.2), (2.2)*(2.2), (-4.8)*(-4.8) = 0.64, 4.84, 1.44, 4.84, 23.04
(4) Get the sum of all the squares in step #3 so that we have:
sum of squares = 0.64 + 4.84 + 1.44 + 4.84 + 23.04 = 34.8
(5) We divide the sum in step #4 by the sample size - 1
34.8/(5-1) = 34.8/4 = 8.7
(6) Last, we take the square root of the value in step #5.
Standard Deviation = sqrt(8.7) = roughly 2.95

Interpreting Standard Deviation.

A smaller standard deviation means that there is more agreement between the numbers (less variation)and a larger standard deviation means that there is less agreement (more variation).
If the observations are random and fall in a bell curve, then we can use the standard deviation to make the following observations:

68% of the numbers lie within one standard deviations of the mean
95% of the numbers lie within two standard deviations of the mean

Now, movie ratings are, in theory, not random since they are based on the quality of a movie. Additionally, we can know that 100% are between 1 and 10 and are most likely whole numbers.
But, what would it say for another movie if the mean were 5 and the standard deviation was 1 and we assume that ratings form a bell curve.
With this information, we can expect:

68% of all people will rate the movie between 4 and 6 since 4= 5-1 and 6 = 5+1
95% of all people will rate the movie between 3 and 7 since 3 = 5 - 2*1 and 7 = 5 + 2*1

PROBABILITY CALCULATIONS --- By. Japhet Masatu.

HOW TO CALCULATE PROBABILITY.

INTRODUCTION:-

Probability is the measure of how an event is likely to occur out of the number of possible outcomes. Calculating probabilities allows you to use logic and reason even with some degree of uncertainty. Find out how you can do the math when you calculate probabilities. Part 1 of 4: Calculating the Probability of a Single Random Event.

1
Define your events and outcomes. Probability is the likelihood of one or more events happening divided by the number of possible outcomes. So, let's say you're trying to find the likelihood of rolling a three on a six-sided die. "Rolling a three" is the event, and since we know that a six-sided die can land any one of six numbers, the number of outcomes is six. Here are two more examples to help you get oriented:
- Example 1: What is the likelihood of choosing a day that falls on the weekend when randomly picking a day of the week?
  - "Choosing a day that falls on the weekend" is our event, and the number of outcomes is the total number of days in a week, seven.
- Example 2: A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is red?
  - "Choosing a red marble" is our event, and the number of outcomes is the total number of marbles in the jar, 20.
2
Divide the number of events by the number of possible outcomes. This will give us the probability of a single event occurring. In the case of rolling a three on a die, the number of events is one (there's only one three on each die), and the number of outcomes is six. You can also think of this as 1 ÷ 6, 1/6, .166, or 16.6%. Here's how you find the probability of our remaining examples:
- Example 1: What is the likelihood of choosing a day that falls on the weekend when randomly picking a day of the week?
  - The number of events is two (since two days out of the week are weekends), and the number of outcomes is seven. The probability is 2 ÷ 7 = 2/7 or .285 or 28.5%.
- Example 2: A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is red?
  - The number of events is five (since there are five total marbles), and the number of outcomes is 20. The probability is 5 ÷ 20 = 1/4 or .25 or 25%. Part 2 of 4: Calculating the Probability of Multiple Random Events.

1
Break the problem into parts. Calculating the probability of multiple events is a matter of breaking the problem down into separate probabilities. Here are three examples:
- Example 1: What is the probability of rolling two consecutive fives on a six-sided die?
  - You know that the probability of rolling one five is 1/6, and the probability of rolling another five with the same die is also 1/6.
  - These are independent events, because what you roll the first time does not affect what happens the second time; you can roll a 3, and then roll a 3 again.
- Example 2:Two cards are drawn randomly from a deck of cards. What is the likelihood that both cards are clubs?
  - The likelihood that the first card is a club is 13/52, or 1/4. (There are 13 clubs in every deck of cards.) Now, the likelihood that the second card is a club is 12/51.
  - You are measuring the probability of dependent events. This is because what you do the first time affects the second; if you draw a 3 of clubs and don't put it back, there will be one less fewer club and one less card in the deck (51 instead of 52).
- Example 3: A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If three marbles are drawn from the jar at random, what is the probability that the first marble is red, the second marble is blue, and the third is white?
  - The probability that the first marble is red is 5/20, or 1/4. The probability of the second marble being blue is 4/19, since we have one fewer marble, but not one fewer blue marble. And the probability that the third marble is white is 11/18, because we've already chosen two marbles. This is another measure of a dependent event.
2
Multiply the probability of each event by one another. This will give you the probability of multiple events occurring one after another. Here's what you can do:
- Example 1:What is the probability of rolling two consecutive fives on a six-sided die? The probability of both independent events is 1/6.
  - This gives us 1/6 x 1/6 = 1/36 or .027 or 2.7%.
- Example 2: Two cards are drawn randomly from a deck of cards. What is the likelihood that both cards are clubs?
  - The probability of the first event happening is 13/52. The probability of the second event happening is 12/51. The probability is 13/52 x 12/51 = 12/204 or 1/17 or 5.8%.
- Example 3: A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If three marbles are drawn from the jar at random, what is the probability that the first marble is red, the second marble is blue, and the third is white?
  - The probability of the first event is 5/20. The probability of the second event is 4/19. And the probability of the third event is 11/18. The probability is 5/20 x 4/19 x 11/18 = 44/1368 or 3.2%.

EditPart 3 of 4: Converting Odds to Probabilities

1
Determine the odds. For example, a golfer is favorite to win at a 9/4 odds. The odds of an event is the ratio of its probability that it will will occur to the probability that it will not occur.
- In the example of 9:4 ratio, 9 represents the probability that the golfer will win. 4 represents the probability he will not win. Therefore, it is more likely for him to win than to lose.
- Remember that in sports betting and bookmaking, odds are expressed as "odds against," which means that the odds of an event not happening are written first, and the odds of an events not happening come second. Although it can be confusing, it's important to know this. For the purposes of this article, we will not use odds against.
2
Convert the odds to probability. Converting odds is pretty simple. Break the odds into two separate events, plus the number of total outcomes.
- The event that the golfer will win is 9; the event that the golfer will lose is 4. The total outcomes is 9 + 4, or 13.
- Now the calculation is the same as calculating the probability of a single event.
  - 9 ÷ 13 = .692 or 69.2%. The probability of the golfer winning is 9/13.

EditPart 4 of 4: Knowing the Probability Rules

1
Ensure that two events or outcomes must be mutually exclusive. That means they both cannot occur at the same time.
2
Assign a probability that is a non-negative number. If you arrive at a negative number, check your calculations again.
3
The likelihood of all possible events needs to add up to 1 or 100%. If the likelihood of all possible events doesn't add up to 1 or 100%, you've made a mistake because you've left out a possible event.
- The likelihood of rolling a three on a six-sided die is 1/6. But the probability of rolling all five other numbers on a die is also 1/6. 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 or 1 or 100%.
4
Represent probability of an impossible outcome with a 0. This just means that there is no chance of an event happening.

TIPS:

You can assign any numbers to events, but they have to be proper probabilities, which means following the basic rules that apply to all probabilities.

You can make your own subjective probability that is based on your opinions about the likelihood of an event. Subjective interpretation of probability will be different for each person.

INTRODUCTION TO PROBABILITY ----- BY. MWL. JAPHET MASATU.

PROBABILITY---By. Mwl. Japhet Masatu.

What is probability?
Probability is the likelihood of something happening in the future.

How is it expressed?

Probability is expressed as a fraction: the denominator is the total number of ways things can occur and the numerator is the number of things that you are hoping will occur:

NUMBER OF THINGS YOU ARE LOOKING FOR PROBABILITY = --------------------------------------- TOTAL NUMBER OF THINGS

Example

Problem

Answer

What is the probability of
a flipped coin turning up heads?

because there is 1 head and the coin has 2 sides.

What is the probability of pulling,
at random, the ace of spades
out of a deck of 52 cards?

because there is only 1 ace of spades and there are 52 cards in the deck.

What is the probability of drawing
any ace out of a deck of 52 cards?

because there 4 aces in a deck of 52 cards.

What is the probability of drawing a red marble out of a bag containing 2 red marbles and 5 blue marbles?

because there are 2 red marbles and 7 marbles in all.

Remember, probability is not dependent on what has happened in the past. So if you have tossed a coin and it has come up heads 6 times in a row, the probability of it coming up heads a 7th time is still

Probability is a branch of mathematics that deals with calculating the likelihood of a given event's occurrence, which is expressed as a number between 1 and 0. An event with a probability of 1 can be considered a certainty: for example, the probability of a coin toss resulting in either "heads" or "tails" is 1, because there are no other options, assuming the coin lands flat. An event with a probability of .5 can be considered to have equal odds of occurring or not occurring: for example, the probability of a coin toss resulting in "heads" is .5, because the toss is equally as likely to result in "tails." An event with a probability of 0 can be considered an impossibility: for example, the probability that the coin will land (flat) without either side facing up is 0, because either "heads" or "tails" must be facing up. A little paradoxical, probability theory applies precise calculations to quantify uncertain measures of random events.
In its simplest form, probability can be expressed mathematically as: the number of occurrences of a targeted event divided by the number of occurrences plus the number of failures of occurrences (this adds up to the total of possible outcomes):

p(a) = p(a)/[p(a) + p(b)]

Calculating probabilities in a situation like a coin toss is straightforward, because the outcomes are mutually exclusive: either one event or the other must occur. Each coin toss is an independent event; the outcome of one trial has no effect on subsequent ones. No matter how many consecutive times one side lands facing up, the probability that it will do so at the next toss is always .5 (50-50). The mistaken idea that a number of consecutive results (six "heads" for example) makes it more likely that the next toss will result in a "tails" is known as the gambler's fallacy , one that has led to the downfall of many a bettor.
Probability theory had its start in the 17th century, when two French mathematicians, Blaise Pascal and Pierre de Fermat carried on a correspondence discussing mathematical problems dealing with games of chance. Contemporary applications of probability theory run the gamut of human inquiry, and include aspects of computer programming, astrophysics, music, weather prediction, and medicine.

Probability

Introduction
Probability is the likelihood or chance of an event occurring.

Probability =	the number of ways of achieving success
	the total number of possible outcomes

For example, the probability of flipping a coin and it being heads is ½, because there is 1 way of getting a head and the total number of possible outcomes is 2 (a head or tail). We write P(heads) = ½ .

The probability of something which is certain to happen is 1.
The probability of something which is impossible to happen is 0.
The probability of something not happening is 1 minus the probability that it will happen.

This video is an animated guide to probability. Expressing probability as fractions and percentages based on the ratio of the number ways an outcome can happen and the total number of outcomes is explained. Experimental probability and the importance of basing this on a large trial is also covered.

Single Events
Example
There are 6 beads in a bag, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow?
The probability is the number of yellows in the bag divided by the total number of balls, i.e. 2/6 = 1/3.
Example
There is a bag full of coloured balls, red, blue, green and orange. Balls are picked out and replaced. John did this 1000 times and obtained the following results:

Number of blue balls picked out: 300
Number of red balls: 200
Number of green balls: 450
Number of orange balls: 50

a) What is the probability of picking a green ball?
For every 1000 balls picked out, 450 are green. Therefore P(green) = 450/1000 = 0.45
b) If there are 100 balls in the bag, how many of them are likely to be green?
The experiment suggests that 450 out of 1000 balls are green. Therefore, out of 100 balls, 45 are green (using ratios).
Multiple Events
Independent and Dependent Events
Suppose now we consider the probability of 2 events happening. For example, we might throw 2 dice and consider the probability that both are 6's.
We call two events independent if the outcome of one of the events doesn't affect the outcome of another. For example, if we throw two dice, the probability of getting a 6 on the second die is the same, no matter what we get with the first one- it's still 1/6.
On the other hand, suppose we have a bag containing 2 red and 2 blue balls. If we pick 2 balls out of the bag, the probability that the second is blue depends upon what the colour of the first ball picked was. If the first ball was blue, there will be 1 blue and 2 red balls in the bag when we pick the second ball. So the probability of getting a blue is 1/3. However, if the first ball was red, there will be 1 red and 2 blue balls left so the probability the second ball is blue is 2/3. When the probability of one event depends on another, the events are dependent.
Possibility Spaces
When working out what the probability of two things happening is, a probability/ possibility space can be drawn. For example, if you throw two dice, what is the probability that you will get: a) 8, b) 9, c) either 8 or 9?

a) The black blobs indicate the ways of getting 8 (a 2 and a 6, a 3 and a 5, ...). There are 5 different ways. The probability space shows us that when throwing 2 dice, there are 36 different possibilities (36 squares). With 5 of these possibilities, you will get 8. Therefore P(8) = 5/36 .
b) The red blobs indicate the ways of getting 9. There are four ways, therefore P(9) = 4/36 = 1/9.
c) You will get an 8 or 9 in any of the 'blobbed' squares. There are 9 altogether, so P(8 or 9) = 9/36 = 1/4 .
Probability Trees
Another way of representing 2 or more events is on a probability tree.
Example
There are 3 balls in a bag: red, yellow and blue. One ball is picked out, and not replaced, and then another ball is picked out.

The first ball can be red, yellow or blue. The probability is 1/3 for each of these. If a red ball is picked out, there will be two balls left, a yellow and blue. The probability the second ball will be yellow is 1/2 and the probability the second ball will be blue is 1/2. The same logic can be applied to the cases of when a yellow or blue ball is picked out first.
In this example, the question states that the ball is not replaced. If it was, the probability of picking a red ball (etc.) the second time will be the same as the first (i.e. 1/3).
This video shows examples of using probability trees to work out the overall probability of a series of events are shown. Both independent and conditional probability are covered.

The AND and OR rules (HIGHER TIER)
In the above example, the probability of picking a red first is 1/3 and a yellow second is 1/2. The probability that a red AND then a yellow will be picked is 1/3 × 1/2 = 1/6 (this is shown at the end of the branch). The rule is:

If two events A and B are independent (this means that one event does not depend on the other), then the probability of both A and B occurring is found by multiplying the probability of A occurring by the probability of B occurring.

The probability of picking a red OR yellow first is 1/3 + 1/3 = 2/3. The rule is:

If we have two events A and B and it isn't possible for both events to occur, then the probability of A or B occuring is the probability of A occurring + the probability of B occurring.

On a probability tree, when moving from left to right we multiply and when moving down we add.
Example
What is the probability of getting a yellow and a red in any order?

Probability

Introduction
Probability is the likelihood or chance of an event occurring.

Probability =	the number of ways of achieving success
	the total number of possible outcomes

The probability of something which is certain to happen is 1.
The probability of something which is impossible to happen is 0.
The probability of something not happening is 1 minus the probability that it will happen.

Number of blue balls picked out: 300
Number of red balls: 200
Number of green balls: 450
Number of orange balls: 50

If two events A and B are independent (this means that one event does not depend on the other), then the probability of both A and B occurring is found by multiplying the probability of A occurring by the probability of B occurring.

The probability of picking a red OR yellow first is 1/3 + 1/3 = 2/3. The rule is:

If we have two events A and B and it isn't possible for both events to occur, then the probability of A or B occuring is the probability of A occurring + the probability of B occurring.

On a probability tree, when moving from left to right we multiply and when moving down we add.
Example
What is the probability of getting a yellow and a red in any order?
This is the same as: what is the probability of getting a yellow AND a red OR a red AND a yellow.
P(yellow and red) = 1/3 × 1/2 = 1/6
P(red and yellow) = 1/3 × 1/2 = 1/6
P(yellow and red or red and yellow) = 1/6 + 1/6 = 1/3

- See more at: http://www.mathsrevision.net/gcse-maths-revision/statistics-handling-data/probability#sthash.iUNOLX5Z.dpuf

This is the same as: what is the probability of getting a yellow AND a red OR a red AND a yellow.

Probability

Introduction
Probability is the likelihood or chance of an event occurring.

Probability =	the number of ways of achieving success
	the total number of possible outcomes

The probability of something which is certain to happen is 1.
The probability of something which is impossible to happen is 0.
The probability of something not happening is 1 minus the probability that it will happen.

Number of blue balls picked out: 300
Number of red balls: 200
Number of green balls: 450
Number of orange balls: 50

If two events A and B are independent (this means that one event does not depend on the other), then the probability of both A and B occurring is found by multiplying the probability of A occurring by the probability of B occurring.

The probability of picking a red OR yellow first is 1/3 + 1/3 = 2/3. The rule is:

If we have two events A and B and it isn't possible for both events to occur, then the probability of A or B occuring is the probability of A occurring + the probability of B occurring.

- See more at: http://www.mathsrevision.net/gcse-maths-revision/statistics-handling-data/probability#sthash.iUNOLX5Z.dpuf

P(yellow and red) = 1/3 × 1/2 = 1/6
P(red and yellow) = 1/3 × 1/2 = 1/6
P(yellow and red or red and yellow) = 1/6 + 1/6 = 1/3

- See more at: http://www.mathsrevision.net/gcse-maths-revision/statistics-handling-data/probability#sthash.iUNOLX5Z.dpuf

Probability

Introduction
Probability is the likelihood or chance of an event occurring.

Probability =	the number of ways of achieving success
	the total number of possible outcomes

The probability of something which is certain to happen is 1.
The probability of something which is impossible to happen is 0.
The probability of something not happening is 1 minus the probability that it will happen.

Number of blue balls picked out: 300
Number of red balls: 200
Number of green balls: 450
Number of orange balls: 50

If two events A and B are independent (this means that one event does not depend on the other), then the probability of both A and B occurring is found by multiplying the probability of A occurring by the probability of B occurring.

The probability of picking a red OR yellow first is 1/3 + 1/3 = 2/3. The rule is:

If we have two events A and B and it isn't possible for both events to occur, then the probability of A or B occuring is the probability of A occurring + the probability of B occurring.

On a probability tree, when moving from left to right we multiply and when moving down we add.

Probability

Introduction
Probability is the likelihood or chance of an event occurring.

Probability =	the number of ways of achieving success
	the total number of possible outcomes

The probability of something which is certain to happen is 1.
The probability of something which is impossible to happen is 0.
The probability of something not happening is 1 minus the probability that it will happen.

Number of blue balls picked out: 300
Number of red balls: 200
Number of green balls: 450
Number of orange balls: 50

If two events A and B are independent (this means that one event does not depend on the other), then the probability of both A and B occurring is found by multiplying the probability of A occurring by the probability of B occurring.

The probability of picking a red OR yellow first is 1/3 + 1/3 = 2/3. The rule is:

If we have two events A and B and it isn't possible for both events to occur, then the probability of A or B occuring is the probability of A occurring + the probability of B occurring.

- See more at: http://www.mathsrevision.net/gcse-maths-revision/statistics-handling-data/probability#sthash.iUNOLX5Z.dpuf

Example
What is the probability of getting a yellow and a red in any order?

P(red and yellow) = 1/3 × 1/2 = 1/6

Probability

Introduction
Probability is the likelihood or chance of an event occurring.

Probability =	the number of ways of achieving success
	the total number of possible outcomes

The probability of something which is certain to happen is 1.
The probability of something which is impossible to happen is 0.
The probability of something not happening is 1 minus the probability that it will happen.

Number of blue balls picked out: 300
Number of red balls: 200
Number of green balls: 450
Number of orange balls: 50

If two events A and B are independent (this means that one event does not depend on the other), then the probability of both A and B occurring is found by multiplying the probability of A occurring by the probability of B occurring.

The probability of picking a red OR yellow first is 1/3 + 1/3 = 2/3. The rule is:

If we have two events A and B and it isn't possible for both events to occur, then the probability of A or B occuring is the probability of A occurring + the probability of B occurring.

- See more at: http://www.mathsrevision.net/gcse-maths-revision/statistics-handling-data/probability#sthash.iUNOLX5Z.dpuf

P(yellow and red or red and yellow) = 1/6 + 1/6 = 1/3

- See more at: http://www.mathsrevision.net/gcse-maths-revision/statistics-handling-data/probability#sthash.iUNOLX5Z.dpuf