5.2B Multiplication Rules

Independent EventsDependent Events

General Multiplication Rule

Independent and Dependent Events

Independent Events: Two events are independent if

knowing that one will occur (or has occurred)

does not change the probability that the

otheroccurs.

Independent and Dependent Events

Dependent Events: Two events are dependent if knowing

that one will occur (or has occurred)

changes the probability that the other occurs.

Example #1

The following are examples of independent

events:a. Rolling a die AND getting a 6, and

then rolling a second die and getting a 3.

b. Drawing a card from a deck AND getting a queen, replacing it, then drawing a second card and

getting a king.

Example #1

The following are examples of independent

events:c. Being on time to school AND your

teacher being on time to school.d. Choosing a marble from a jar

AND tossing a coin that lands on heads.

Example #2

The following are examples of dependent

events:a. The speed you drive to school

AND the weather.b. Choosing a marble from a jar,

not replacing it, AND drawing another marble from that same jar..

Example #2

The following are examples of dependent

events:c. Eating a full breakfast AND

being on time to school.d. Parking in a no-parking zone

AND getting a parking ticket.

Determine whether the events are independent or dependent.a.Tossing a coin and drawing a marble

out of a bag.INDEPENDENT

b.Eating sweets and having diabetes.DEPENDENT

Example #3

Example #3

Determine if the events are independent or

dependent.c.Being on the Indianapolis Colts football

team and being a winnerDEPENDENT

d.Drawing a king from a standard deck, replacing it and drawing another king.INDEPENDENT

Multiplication Rule For Independent Events

If events A and B are independent,

BPAPBandAP

Example #4

A dresser drawer contains one pair of socks of each of the following colors: blue, brown, red, white and black. Each pair is folded together in matching pairs. You reach into the sock drawer and choose a pair of socks without looking. The first pair you pull out is red -the wrong color.

You replace this pair and choose another pair. What

is the probability that you will choose the red pair of socks twice?

Example #4

Indepdendent?Yes

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Example #5

A coin is tossed and a single 6-sided die is

rolled. Find the probability of landing on the head side of the coin and rolling a 3

on the die. Independent?Yes

Example #5

33 PHPandHP

083.12

1

6

1

2

1

Example #6

A card is chosen at random from a deck of

52 cards. It is then replaced and a second

card is chosen. What is the probability of choosing a face card and an eight? Independent?Yes

Example #6

88 PFacePandFaceP

018.52

4

52

12

Example #7

A South Carolina survey of registered voters

found that 65% were opposed to the new Health Care Plan. Suppose you randomly choose 5 South Carolinians. What is the probability all 5 of them oppose the health care plan?Independent?Yes

Example #7

OandOandOPopposeP 3

275.65.65.65.65. 3

General Multiplication Rule

Given events A and B, the probability of

both A and B occurring is:P(A and B) = P(A)P(B|A), Where P(B|A) is the probability that B

occurs given A has occurred.

Example #8

A card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a second card is chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack? Independent?No

Example #8

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4

52

4

Example #9

Mr. Parietti needs two students to help him with a science demonstration for his class of 18 girls and 12 boys. He randomly chooses one student who comes to the front of the room. He then chooses a second student from those still seated. What is the probability that both students chosen are girls? Independent?No

Example #9

GGPGPGP |2

152.29

11

30

12

Example #10

In a shipment of 20 computers, 3 are defective. Three computers are randomly selected and tested. What is the probability that all three are defective if the first and second ones are not replaced after being tested? Independent?No

Example #10

DDPDDPDPDefectiveP 2|1|3

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1

19

2

20

3

Example #11

On a math test, 5 out of 20 students got an A. If three students are chosen at random without replacement, what is the probability that all three got an A on thetest? Independent?No

Example #11

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3

19

4

20

5

Example #12

A jar contains 6 red balls, 3 green balls, 5 white balls and 7 yellow balls. Two balls are chosen from the jar, with replacement. What is the probability that both balls chosen are green? Independent?Yes

Example #12

GPGPGP 2

0204.21

3

21

3

Example #13

A nationwide survey showed that 73% of all children in the United States dislike eating vegetables. If 5 children are chosen at random, what is the probability that all 5 dislike eating vegetables?Independent?Yes

Example #13

55 VeggiesDislikePVeggiesDislikeP

207.73. 5

Example #14

A school survey found that 7 out of 30 students walk to school. If four students are selected at random without replacement, what is the probability that the first two chosen walk to school and the next two do not walk to school? Independent?No

Example #14

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