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# Conditional probability and the multiplication rule

## 1. Conditional Probability and the Multiplication Rule

Unit 3.3CONDITIONAL PROBABILITY AND

THE MULTIPLICATION RULE

## 2. Multiplication Rule

The Multiplication Rule can be used to findthe probability of two or more events that

occur in a sequence .

The multiplication Rule for the probability of

A and B

If events A and B are independent, then the rule

can be simplified to P(A and B) = P (A) ● P (B). This

simplified rule can be extended for any number of

independent events.

## 3. Multiplication Rule Tip

1. Find the probability the first event occurs.2. Find the probability the second event occurs

given the first event has occurred and

3. Multiply these two probabilities.

## 4. Using the Multiplication Rule to find Probability

1. A coin is tossed and a die is rolled. Find theprobability of getting a head and then rolling

a 6.

## 5. Using the Multiplication Rule to find Probability

1. A card is drawn from a deck and replaced;then a second card is drawn. Find the

probability of selecting a Ace and then

selecting a queen.

## 6. Using the Multiplication Rule to find Probability

1. The probability that a salmon swimssuccessfully through a dam is 0.85. Find the

probability that two salmon successfully

swim through the dam.

## 7. Using the Multiplication Rule to find Probability

1. Two cards are selected from a standard deckwithout replacement. Find the probability

that both are hearts.

## 8. Using the Multiplication Rule to find Probability

1. A Harris poll found the 46% of Americanssay they suffer great stress at least once a

week. If three people are selected at

random, find the probability that all three

will say they suffer great stress at least once

a week.

## 9. Using the Multiplication Rule to find Probability

1. The probability that a salmon swimssuccessfully through a dam is 0.85. Find the

probability that three salmon swim

successfully through the dam.

## 10. Using the Multiplication Rule to find Probability

1. Find the probability that none of the threesalmon are successful.

## 11. Using the Multiplication Rule to find Probability

1. Find the probability that at least one of thethree salmon is successful in swimming

through the dam.

## 12. Dependent Events

When the outcome or occurrence of the firstevent affects the outcome or occurrence of

the second event in such a way that the

probability is changed, the events are said to

be dependent events.

Examples

Drawing a card from a deck, NOT replacing it, and

then drawing a second card.

Being a lifeguard and getting a tan.

Having high grades and getting a scholarship

## 13. Conditional Probability

To find probabilities when events aredependent, use the multiplication rule with a

The probability of B

modification in notation.

given that event A

P(A and B) = P (A) ● P (B\A). has already occured

## 14. Finding Conditional Probability

Two cards are selected in sequence from astandard deck. Find the probability that the

second card is a queen, given that the first

card is a king. (Assume that the king is not

replaced)

Solution: Because the first card is a king and

is not replaced, the remaining deck has 51

cards, 4 of which are queens. So,

P(B|A) = 4/51 = 0.078

## 15. Finding Conditional Probability

Three cards are drawn from an ordinary deckand not replaced. Find the probability of

these events.

Getting 3 Jacks

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

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

Getting three clubs

## 16. Finding Conditional Probability

GenePresent

Gene

Not

Present

Total

High IQ

33

19

52

Normal

IQ

39

11

50

Total

72

30

102

The table at the left shows the

results of a study in which

researchers examined a

child’s IQ and the presence of

a specific gene in the child.

Find the probability that a

child has a high IQ given that

the child has the gene.

Solution: There are 72

children who have the gene.

So, the sample space consists

of these 72 children, as shown

at the left. Of theses, 33 have a

high IQ. So,

P(B\A)= 33/72 =0.458

## 17. Finding Conditional Probability

GenePresent

Gene

Not

Present

Total

High IQ

33

19

52

Normal

IQ

39

11

50

Total

72

30

102

1. Find the probability

that a child does not

have the gene.

2. Find the probability

that a child does not

have the gene, given

that the child has a

normal IQ.