9.7 Probability

Any happening whose result is uncertain is called an experiment, and its results are called outcomes. The experiment's sample space is the collection of every possible outcome it has, and an event is any subcollection of the sample space.

Probability

If an event has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes, the probability of the event is:

 

Example 1

 A card is drawn from a standard 52 card deck. What is the probability that the card is a red 7?

n(E) = 2 cards (7 of hearts and 7 of diamonds)

n(S) = 52 cards

= 2/52 = 1/26 

Mutually Exclusive Events

Two events are mutually exclusive if they have no outcomes in common. The intersection of two sets A and B is . If A and B are mutually exclusive, then  = 0. 

If A and B are events in the same sample space, the probability of A or B occurring is 

P(A U B) = P(A) + P(B) -    

If A and B are mutually exclusive, then 

P(A U B) = P(A) + P(B)

Example 2

One card is selected from a deck of 52 playing cards. What is the probability that the card is either a spade (A) or a 3 (B)?   

P(A) = 13/52

P(B) = 4/52

  = 1/52

P(A U B) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13

Independent Events 

Two events are independent if the occurrence of one has no effect on the occurrence of the other. To find the probability of both events happening, multiply the probabilities of each together. 

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

Example 3

One bag contains 5 blue marbles and 1 white marble, and the other contains 4 red marbles and 5 white marbles. A marble is taken from the first bag A, and then a marble is taken from the second bag B. What is the probability of both marbles being white?

P(A) = 1/6

P(B) = 5/9

P(A and B) = (1/6) * (5/9)  = 5/54

The Complement of an Event 

The complement of an event is the collection of all the outcomes in the sample space that are not in the event. The complement of event A is denoted by A'. P(A') = 1 - P(A).