Thinking Mathematically

Counting Methods and Probability11.4 Fundamentals of Probability

Computing Theoretical Probability

Sspacesampleinoutcomesofnumber

EeventinoutcomesofnumberEP )(

If an event E has n(E) equally-likely outcomesand its sample space S has n(S) equally-likelyoutcomes, the theoretical probability of event E denoted by P(E), is

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.

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

More Examples

Exercise 11.4 #5, 21, 29, 37A 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?

Thinking Mathematically

Counting Methods and Probability11.5 Probability with the Fundamental Counting

Principle, Permutations, and Combinations

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 thata. Comic E will perform first

b. Comic C will perform fifth and comic B will perform last.

c. The comedians will perform in the following order D, E, C, A, B, F.

d. Comic A or comic B will perform first.

Example: Combinations

Exercise 11.5 # 5, #17

A group consists of four men and five women. Three people are selected to attend a conference.

a. In how many ways can three people be selected from this group of nine?

b. In how many ways can three women be selected from the five women?

c. 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.

Thinking Mathematically

Counting Methods and Probability11.6 Events Involving Not and Or

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 #1If you are dealt one card from a deck of 52, what is the

probability that the card is not an ace.

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, thenP(A or B) = P(A) + P(B).

Exercise 11.6 #17One card is randomly selected from a deck of 52. What is the probability that the card is either a 2 or a 3?

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 #33You 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?

Thinking Mathematically

Counting Methods and Probability11.7 Events Involving And; Conditional Probability

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)

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.

Dependent Events

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

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.

Example: And with Dependent Events

Exercise 11.7 #43An 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?

Thinking Mathematically

Chapter 11

Counting Methods and Probability