7.7 - Probability

If you enjoy this section, take a Finite or Statistics course. Take a Finite or Statistics course even if you don't enjoy this section, they're loads of fun.


Any happening whose result is uncertain.
Possible results from an experiment
Sample Space
Set of all possible outcomes
Subset of the sample space. One or more outcomes.
Equally Likely Events
Events which have the same chance of occurring
Chance that an event will occur. Theoretically for equally likely events, it is the number of ways an event can occur divided by number of outcomes in the sample space. Empirically, the long term relative frequency.
Independent Events
Events in which the occurrence of one event does not change the probability of the occurrence of the other. One does not affect the other.
Dependent Events
Events that are not independent.
Mutually Exclusive Events
Events that can not happen at the same time. Disjoint events.
All Inclusive Events
Events whose union comprises the totality of the sample space.
Complementary Events
Two mutually exclusive events that are all inclusive.

Sample Spaces

The sample space is the set of all the possible outcomes in an experiment and is denoted by a capital letter S.

If you were to roll a single die, then S = { 1, 2, 3, 4, 5, 6 }, the set of all possible outcomes.

If you were to roll two dice and look at the sum of the two dice, then S = { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }.

Equally Likely Events

However, not all sample spaces are created equally. In fact, that last example is not. There is only one way that a sum of 2 can be rolled, a 1 on the first die and a 1 on the second die. There are four ways that a sum of five can be rolled: 1-4, 2-3, 3-2, 4-1 (don't be confused here, 1-4 is a 1 on the first die and a 4 on the second and is different than a 4 on the first and a 1 on the second. If it helps, pretend that you're rolling one die and a friend is rolling the other).

We want our sample spaces to be equally likely if at all possible.

Classical / Theoretical Probability

If outcomes are equally likely, then the probability of an event occurring is the number in the event divided by the number in the sample space.

P(E) = n(E) / n(S)

The probability of rolling a six on a single roll of a die is 1/6 because there is only 1 way to roll a six out of 6 ways it could be rolled.

The probability of getting a sum of 5 when rolling two dice is 4/36 = 1/9 because there are 4 ways to get a five and there are 36 ways to roll the dice (Fundamental Counting Principle - 6 ways to roll the first times 6 ways to roll the second).

Do not make the mistake of saying that the probability of rolling a sum of 5 is 1/11 because there is one 5 out of a sample space of 11 sums (2 through 12). When the sample spaces are not equally likely, do not divide by the number in the sample space.

Properties of Probabilities

Addition Rules

When you want to find the probability of one event OR another occurring, you add their probabilities together.

This can lead to problems however, if they have something in common.

The probability of one or both of two events occurring is ...

P(A or B) = P(A) + P(B) - P(A and B)

Mutually Exclusive Events

Mutually Exclusive Events have nothing in common. The intersection of the two events is the empty set. The probability of A and B both occurring is 0 because they can't occur at the same time.

If two events are mutually exclusive, then the probability of one or the other occurring is ...

P(A or B) = P(A) + P(B)

Multiplication Rules

When you want to find the probability of two events both occurring, then you need to apply the Fundamental Counting Principle. This principle can be extended to probabilities.

Independent Events

Independent Events are events where one occurring doesn't change the probability of the other occurring. When events are independent, the probability of them both occurring is ...

P(A and B) = P(A) * P(B)

We don't have time to get into probability very deeply. If we did, we would cover conditional probability - the probability of dependent events.

Complementary Events

The root word in complementary is "complete". Complementary events complete, or make whole. Complementary events are mutually exclusive, but when combined make the entire sample space.

The symbol for the complement of event A is A'. Some books will put a bar over the set to indicate its complement.

Since complementary events are mutually exclusive, we can use the special addition rule to find its probability. Furthermore, complementary events are all inclusive, so they make the sample space when combined, so their probabilities have a sum of 1.

The sum of the probabilities of complementary events is 1.

P(A) + P(A') = 1
P(A') = 1 - P(A)