Probability Basics

Probability Cheat Sheet

Basics

Probability – The likelihood of an event occurring.

Event – a specific outcome, or a combination of several outcomes.

A → event

P(A) → probability of A occurring

This is also written:

The probability of two independent events occurring at the same time is equal to the product of all the probabilities of the individual events.

Computing Expected Values

Expected Values – the average outcome we expect if we run an experiment many times. We can use Expected Values to make predictions about the future based on past data.

Experimental Probabilities – the probabilities we get after conducting experiments.

Theoretical (or True) Probabilities – calculated probabilities, such as the odds of coin flips or dice rolls. These were demonstrated in the ‘Basics’ section above.

Expected Values for Categorical Outcomes

The Expected Value is the outcome we expect to occur when we run an experiment. The Expected Value of A occurring is written E(A).

where n is the number of trials carried out.

Example: What is the Expected Value of drawing a spade from a full deck of cards if 20 card pulls are performed?

The probability of drawing a space is 13/52 = 0.25 = P(A). And 20 card pulls means n = 20.

That means we expect to draw a spade 5 times when a card is pulled from a full deck 20 times.

Expected Values for Numerical Outcomes

We take the value for every element in the sample space, and multiply it by its probability. Then we add these together:

Frequency of Outcomes

Probability Frequency Distribution – a collection of the probabilities for each possible outcome. This can be represented graphically or in a table.

Example: Let’s look at the sum of rolling two 6-sided dice.

	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

Record the number of times each sum appears in the above table in ascending order. Also record the frequency of each sum. This is a Frequency Distribution Table.

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

In the above example, the sum of ‘8’ has a frequency of ‘5’.

Divide the Frequency by the size of the sample space, in this case, 6² = 36 possible outcomes to get the probability of each event. Add this to the above table and you have the Probability Frequency Distribution:

Sum	Frequency	Probability
2	1	1/36
3	2	1/18
4	3	1/12
5	4	1/9
6	5	5/36
7	6	1/6
8	5	5/36
9	4	1/9
10	3	1/12
11	2	1/18
12	1	1/36

Events and their Complements

Complement – The complement of an event is everything that the event is not, such that A + A’ = sample space, where A’ is the complement.

Note that the complement of a complement is the event itself:

Example: We have a sample space with three possible outcomes: A, B, and C:

Probability Cheat Sheet