1. Thinking Mathematically
Counting Methods and Probability
11.4 Fundamentals of Probability

2. Computing Theoretical Probability
If an event E has n(E) equally-likely outcomes
and its sample space S has n(S) equally-likely
outcomes, the theoretical probability of event
E denoted by P(E), is

3. Example Probability and a Deck of 52 Cards
You are dealt one card from a standard 52-card deck. Find the probability of being dealt a King.

4. Solution
Because there are 52 cards, the total number of possible ways of being dealt a single card is 52. We use 52, the total number of possible outcomes, as the number in the denominator. Because there are 4 kings in the deck, the event of being dealt a king can occur 4 ways.
P(king) = 4/52 = 1/13

5. More Examples Exercise 11.4 #5, 21, 29, 37
A die is rolled, what is the probability of rolling a number less than 3?
A fair coin is tossed two times, what is the probability of getting two heads?
Arbitrarily selecting a family with three children, what is the probability that the family has exactly two male children?
A single die is rolled twice, what is the probability of rolling two numbers such that their sum is 5?

6. Thinking Mathematically
Counting Methods and Probability
11.5 Probability with the Fundamental Counting Principle, Permutations, and Combinations

7. Example: Permutations
Exercise 11.5 #3
Six stand-up comics, A, B, C, D, E, and F are to perform on a single evening. The order of performance is determined randomly. Find the probability that
Comic E will perform first
Comic C will perform fifth and comic B will perform last.
The comedians will perform in the following order D, E, C, A, B, F.
Comic A or comic B will perform first.

8. Example: Combinations
Exercise 11.5 # 5, #17
A group consists of four men and five women. Three people are selected to attend a conference.
In how many ways can three people be selected from this group of nine?
In how many ways can three women be selected from the five women?
Find the probability that the selected group will consist of all women.
If you are dealt 4 cards from a shuffled deck of 52, find that probability of getting two queens and two kings.

9. Thinking Mathematically
Counting Methods and Probability
11.6 Events Involving Not and Or

10. The Probability of an Event Not Occurring
The probability that an event E will not occur is equal to 1 minus the probability that it will occur.
P(not E) = 1 - P(E)
Exercise 11.6 #1
If you are dealt one card from a deck of 52, what is the probability that the card is not an ace.

11. Mutually Exclusive Events
If it is impossible for events A and B to occur simultaneously, the events are said to be mutually exclusive.
If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B).
Exercise 11.6 #17
One card is randomly selected from a deck of 52. What is the probability that the card is either a 2 or a 3?

12. Or Probabilities with Events That Are Not Mutually Exclusive
If A and B are not mutually exclusive events, then
P(A or B) = P(A) + P(B) - P(A and B)
Exercise 11.6 #33
You spin a spinner which has 8 equally likely outcomes, numbered 1 through 8. What is the probability that your spin will be either an even number or a number greater than 5?

13. Thinking Mathematically
Counting Methods and Probability
11.7 Events Involving And; Conditional Probability

14. Independent Events
Two events are independent events if the occurrence of either of them has no effect on the probability of the other.
If A and B are independent events, then
P(A and B) = P(A)•P(B)
Compare to P(A or B)

15. Example: And with Independent Events
Exercises 11.7 #3, 13, 19
A spinner has 3 red regions, 2 green regions, and 1 yellow regions. Each of the 6 regions are equally likely. If the spinner is spun twice, what is the probability of two successive yellow spins?
A single die is rolled twice. What is the probability of rolling an even number on the first roll and a number greater than 2 the second time.
A card is drawn from a deck of 52. The card is replaced, the deck shuffled and a second card is drawn. What is the probability of drawing a red card each time.

16. Dependent Events
Two events are dependent events if the occurrence of one of them has an effect on the probability of the other.

17. And Probabilities with Dependent Events
If A and B are dependent events, then
P(A and B) = P(A)•P(B given that A has occurred)
The conditional probability of B, given A, written P(B|A), is the probability that event B occurs computed on the assumption that event A occurs.

18. Example: And with Dependent Events
Exercise 11.7 #43
An ice chest contains six cans of apple juice, eight cans of grape juice, four cans of orange juice, and two cans of mango juice. If you randomly choose three cans in succession, what is the probability of choosing three cans of apple juice?

19. Thinking Mathematically
Chapter 11
Counting Methods and Probability