11.7 Events Involving AND; Conditional Probability

AND Probabilities with Independent Events

Two events are independent events if the occurrence of either of them has no effect on the probability of the other.

Example: Winning the lottery twice; getting three consecutive tails when tossing a coin 3 times; being struck (or almost struck) by a lightning twice.

AND Probabilities with Independent Events

P (A and B) = P (A) x P (B)

Problem 1

Find the probability that a married couple will have 3 baby girls in a row.

1/8

Problem # 2

To play the Powerball, a player needs to select five numbers from 1-59 and a Powerball number from 1-39. Find the probability of winning twice in a row.

2.62 x 10^-17

Probability of an Event Happening at least Once

P (event happening at least once) =

1 – P (event does not happen)

Or1 – P (not)

Problem # 3 (Ex. 3 from text)

If the probability that South Florida will be hit by a hurricane in any single year is 5/19. What is the probability that

A. South FL will be hit by a hurricane in 3 consecutive years?

B. South FL will not be hit by a hurricane in the next ten years?

C. South FL will be hit by a hurricane at least once in the next 10 years?

A. 125/6859 = 0.018

B. 0.047

C. 0.953

Problem # 4

In a poker game, the probability of getting a full house (3 cards of one number and 2 cards of a second number) is 3744 / 2,598,960.

A. Find the probability of getting a full house 3 times in a row.

B. Find the probability of not getting a full house after 20 games.

C. Find the probability of getting a full house at least once in 20 games.

A. 2.99 x 10^-9

B. 0.97

C. 0.03

AND Probabilities with Dependent Events

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

Example: You are dealt two cards from a 52-deck of card. Find the probability of getting 2 Aces.

P (A and B) = P (A) x P (B given that A has occurred)

Problem # 5

There are 20 chocolates in a box and all are identically shaped. 5 are solid chocolate, 8 are filled with caramel, 4 are filled with peanuts and 3 are filled with coconut. Find the probability of getting

A. 2 filled with caramels in a row

B. Solid chocolate followed by one filled with peanut

A. 0.15

B. .05

Problem 6 (Ex. 5 from text)

Three people are randomly selected, one person at a time, from 5 freshmen, 2 sophomores and 4 juniors. Find the probability that the first 2 people selected are freshmen and the 3rd is a junior.

8/99

Conditional Probability

The probability of event B, assuming that the event A has already occurred, is called conditional probability of B, given A. The probability is denoted by

P (B A)

Problem 7 (Ex 6 from text)

A letter is randomly selected from the letters of the English alphabet. Find the probability of selecting a vowel, given that the outcome is a letter that precedes h.

2/7

Problem 8 (Ex 7 from text)

The table on the left shows the differences in the political ideology per 100 males and 100 females in the 2000 US presidential election. Find the probability that the person

A. is liberal, given that the person is female

B. is male, given that the person is conservative

Liberal Moderate Conservative

M 16 45 39

F 20 55 25