Unit 4Section 3.2

3.2: Conditional Probability and the Multiplication Rule Conditional Probability (of event B)

– probability that event B occurs after event a has already occurred.

Notation: P(B|A)

Formula:

P(B|A): Probability of “B given A”

Example 1: A box contains black and white chips. A person selects two

chips without replacement. If the probability of selecting a black and white chip is 15/56, and the probability of selecting a black chip of the first draw is 3/8, find the probability of selecting a white chip in the second drawn, given the first chip selected was black.

Example 2: The probability that Sam parks in a no-parking zone and

gets a parking ticket is 0.06, and the probability that Sam cannot find a legal parking space and has to park in the no-parking zone is 0.20. On Tuesday, Sam arrives at school and has to park in a no-parking zone. Find the probability that he will get a parking ticket.

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Example 3: A recent survey asked 100 people if they thought women

in the armed forces should be permitted to participate in combat. The results are shown below:

Find these probabilities:a) The respondent answered yes, given that the respondent was a female.

b) The respondent was male, given that the respondent answered no.

Gender Yes No Total

Male 32 18

Female 8 42

Total :

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Independent events - two events (A and B) are independent if the fact that A occurs does not affect the probability of B occurring .

EX: rolling a die twice

When two events are independent, the probability of both occurring is:

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

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Example 4: Find the probability of the following.

A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die.

A card is drawn from a deck and replaced; then a second card is drawn. Find the probability of getting a queen and then an ace.

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Example 5: Find the probability of the following.

A jar contains 3 red marbles, 2 blue marbles, and 5 white marbles. A ball is selected then replaced. Find the probability of selecting …

A blue marble, then a second blue marble.

A blue marble, then a white marble.

A red marble, then a blue marble.

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Example 6: A Harris poll found that 46% of Americans say they suffer from great stress at least once a week. If three people are selected at random, find the probability they will say they suffer at least once a week.

Example 7: Approximately 9% of men have a type of color blindness that prevents them from distinguishing between red and green. If three men are selected at random, find the probability that all of them will have this type of color blindness.

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Dependent events – When the outcome of occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probabilities are changed.

EX: Selecting a card from a deck, not replacing it, then selecting another card from a deck.

When two events are dependent, the probability of both occurring is:

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

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Example 8: A person owns a collection of 30 CDs, of which 5 are

country music. If 2 CDs are selected at random, find the probability that both are country music.

Example 9: Three cards are drawn from an ordinary deck and not replaced. Find the probability of the following:

Getting 3 jacks

Getting an ace, a king, and a queen in order

Getting a club, a spade, and a heart in order

Getting 3 clubs

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Example 10: Box A contains 2 red marbles and

1 white marble. Box B contains 3 blue marbles and 1 red marble. A coin is tossed. If it falls heads up, Box A is

selected and a ball is drawn. If it falls tails up, Box B is selected and a ball is drawn. Find the probability of selecting a red ball.

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Probabilities for “at least”

In order to determine probabilities involving “at least,” we calculate the probability using its complement.

For example:

When drawing 4 cards from a deck, to find the probability of at least one of them being an ace...we would determine the probability of selecting NO aces, then subtract the value from 1.

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Example 11: A coin is tossed 5 times. Find the probability

of getting at least one tail.

Example 12: The Neckware Associates of America

reported that 3% of tires sold in the United States are bowties. If 4 customers who purchased a tie are randomly selected, find the probability that at least one purchased a bow tie.

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Homework: Pg 152 - 154: (7 - 27 ODD)

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