Do you like winning at games or taking risks? Persuading others to see your point of view? If you do, you will love statistics and probability. These tools are you mathematics 8-ball that can help you make really intelligent, uncanny predictions.

Here is a course called Workshop on Probability and Statistics that helps clear thing up, but let’s go over the basics on how you can start to learn statistics.

## Computing Probability

You may already have come across probability today. Was there a thirty per cent of rain in the weather forecast? Did you not study for history class because you just had a pop quiz two days ago? These decisions and predictions are based on statistics and probability.  As a matter of fact, probability is related to statistics since most probabilities are based on what occurred in past events. It is important to know where data comes from when you are trying to understand probability. For even more introductory statistics information, here is a course about descriptive statistics you might find useful.

A probability is just fractions that can be written as ratios or percents. A probability’s numerator is the number of outcomes satisfying the probability’s conditions, while the denominator is how many possible outcomes there could possibly be:

Number of outcomes that are favorable/ Number of outcomes possible

For example:

• What is the probability that the sum of two 6-sided dice will be more than ten?

The first step would be to look at all the sums possible that you can get from 2 dice being rolled. Each of the dice has the possible outcomes of: 1-2-3-4-5 and 6. These combinations can be represented in a table where the first die’s possibilities are on the horizontal line and the second die’s possibilities are on the vertical line.

 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12

In the table, the combination of both dice amount to 36 possibilities, three of which are more than ten. Thus, there are three favorable outcomes and thirty-six possible outcomes:

3/36 = 1/12 (simplified) = 8.3%

## Probability Using And/Or

When it comes to computing probability, there are big distinctions between the words “or” and “and.” When you see the word “or, this means that the outcome has to satisfy 1 condition, or the other condition or both conditions at the same time. On the other hand, “and” means that both conditions have to be satisfied by the outcome simultaneously.

For example:

• What is the probability of taking a card from 1 deck and it being a face OR a red card?

When computing for probability with an “or” questions, the card can be a face card, a red card or both simultaneously. The red cards in a deck number twenty-six, six of which are card faces. There are also six more card faces that aren’t red including the Queen of Spades, King of Spades, King of Clubs, Jack of Spades, Jack of Clubs and the Queen of Clubs This is a total of 32 cards (26 + 6). Remember not to simply add up the card faces with the red cards as this gives you 38 as the sum, but counting the red cards two times.

32/52 = 8/13 = 61%

## Mutually Exclusive and Complementary Events

An event is mutually exclusive when it has 2 or more outcomes that cannot occur at the same time. Picking a card from a deck and selecting a king or an ace are mutually exclusive since you can’t do these simultaneously. On the other hand, choosing a king or a red card are not mutually exclusive as you can choose a red king.

An event is complementary when 2 outcomes of an event are the only 2 outcomes possible. Rolling dice and getting 2 or 4 are not complementary since there are other possible outcomes such as 1, 3, 5 etc. However, rolling a dice and getting a 1 or NOT getting a 1 is complementary as there are only 2 choices. One example of a complementary event is flipping a coin, since the only 2 choices are either tails or heads.

All mutually exclusive events are not necessarily complementary but all complementary events are mutually exclusive.

## Observing Versus Predicting Probability

Remember that Matthew Mcconaughey movie called “Two for the Money?” He made a lot of money sports betting by observing and predicting probable outcomes of sports events. Basically, when it comes to probability, there are 2 ways to calculate this, either by actually keeping score as you observe the event or by using math to predict. Here is a course you can use to help you determine the outcome of a hypothesis you have.

Observing probability is also known as Experimental Probability. This is based on observing an experiment or trial, counting the outcomes that are favorable and dividing this by the number of times total that the trial was done. For example, if you toss a coin thirty-six times and recorded each outcome:

H-T-H-H-T-H-H-T-H-H-T-H-H-H-H-T-H-H-T-H-H-T-T-T-T-T-T-T-T-H-T-T-H-H-T-H

Based on this experiment, the probability of flipping tails is 17/36 or about 47% while the probability of heads getting flipped is about 19/36 or 53%.

## Compound Events

Compound events in probability would have to do with the chances of 2 or more events occurring at one time. For instance, what are the chances (probabilities) that there will be an exam today and you forgetting your homework? Believe it or not, probabilities such as this can be computed.  Here is an article you might like that is entitled Correlation Coefficient Formula: Statistics Made Easy which gives you more tools to come up with the right figures.

## Using Tree Diagrams to Measure Compound Events

You get the same answer as organized lists when you use a tree diagram. For the die and coin example again, what is the probability of getting an even number and tails?

Make a tree to chart all the outcomes possible. The first ‘branches’ set will be all the outcomes possible for the 1st event. You can then draw branches from each of these outcomes for every possibility of a second event. It doesn’t matter what event you put first, the outcomes total will be the same as the organized lists.

When you count the smallest branches, you will see that there are twelve possibilities so the probability of rolling an even number and flipping a tail is:

3/12 = 1/4 = 25%

## Use Organized Lists to Measure Compound Events

Using the method of organized lists, all the outcomes possible that may occur should be listed down. This is sometimes hard since most people forget 1 or 2 options.

If you toss 1 coin 3 times, what is the probability of 2 heads getting flipped?

In this example, you are working with 3 different events, as each flip equals one occurrence). For things to be kept organized, you can list this in 3 columns:

 Flip 1 Flip 2 Flip 3 T T T T T H T H T T H H H T T H T H H H T H H H

The different outcomes total 8, and of these the favorable ones are HHH, THH, HTH and HHT. So the probability is:

4/8 = 50%

Also, if you start rolling a die and flip a coin, what are the probabilities of getting even numbers and tails?

Begin by making a list of all the outcomes possible that you can get (T1 means getting a 1 and flipping heads):

 T1 H1 T2 H2 T3 H3 T4 H4 T5 H5 T6 H6

The possible outcomes total twelve, with three giving the outcome desired (a number that is even and tails). These are T6, T4 and T2. So:

3/12 = 1/4 = 25%

Here is another example:

• Clara’s closet contained 4 pairs of jeans (blue, grey, white and brown). She also has 5 different tank tops (purple, yellow, red, white and blue). With these options, what various outfits can she create?

Also, if she blindly picked jeans and a tank top, what probability is there that the tank and jeans match?

 Blue-Red Blue-Purple Blue-White Blue-Blue Blue-Blue White-Red White-Purple White-White White-Blue White-Blue Grey-Red Grey-Purple Grey-White Grey-Blue Grey-Blue Brown-Red Brown-Purple Brown-White Brown-Blue Brown-Blue

The first thing you need to do is to get all the different outfits listed. The 1st color will be for jeans and the 2nd color will be for tank tops:

So from this list we see that there are twenty possible outcomes.

There are just 2 matching outfits, (blue-blue and white-white), thus the probabilities of picking matching jeans and a tank top blindly is:

2/20 = 10%

## Factorials

When you need help counting things like order of events or arrangement of items, you get a lot of help when using factorials. Let’s say you want to organize 6 books on your bookshelf. Use factorials to count the different possible methods your books can be organized. In this instance, you could organize your books in “6 factorial” different ways.

The “!” sign is the mathematical sign for factorial. This does not mean you should yell out the number with excitement each time you see it.

Six Factorial

6! = 1 x 2 x 3 x 4 x 5 x 6 =720 so there are seven hundred twenty ways your book can be arranged. Think about it:

There are six options for book 1 on the shelf. You have 5 remaining options for book 2, 4 options for book 3, 3 options for book 4, two options for book5 and one option for book 6.

When you have a factorial, all you really need to do is to multiply all the numbers equal to or less than the number given. For example:

5! = 5 x 4 x 3 x 2 x 1= 120

Back to the book example, if these are arranged randomly, what possibilities are there that they will be alphabetically arranged?

The answer is, there is only 1 way to get them alphabetically arranged, but to arrange them there are 6! =720 so:

1/6! = 1/720 so the answer is: .138%

Does this help? Here is a course on Practical Statistics you can apply to everyday life using statistical questions and helping you determine which tests to use in which situations.

Page Last Updated: February 2020

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