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Independent Events. Independent events are two events in which the occurrence of one has no effect on the probability of the other. Dependent Events. Dependent events are two events in which the occurrence of one changes the probability of the other. Carl. Dick. Heather. Ellen. Alan. Greg.

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slide1

Independent Events

Independent events are two events in which the occurrence of one has no effect on the probability of the other.

slide2

Dependent Events

Dependent events are two events in which the occurrence of one changes the probability of the other.

slide3

Carl

Dick

Heather

Ellen

Alan

Greg

slide4

Probability of Dependent Events

If A and B are dependent events, then P(A and B) = P(A) x P(B|A).

slide5

Probability of Dependent Events

P(B|A) is read as “probability of B given A.”

slide6

Example 1

Select a name from the box and then select a second name without replacing the first. Find the probability of drawing a boy’s name followed by a girl’s name.

slide7

46

23

25

415

=

=

=

=

23

25

= x

B = Select a boy’s name.

G|B = Select a girl’s name, given that a boy’s name was selected on the first draw.

P(B)

P(G|B)

P(B and G)

= P(B) x P(G|B)

≈ 0.27

slide8

Example 2

A bag of chocolate candies contains ten brown, eight orange, three yellow, and four green candies. What is the probability that the first two candies drawn from the bag without replacement will be brown?

slide9

1025

25

38

924

320

=

=

=

=

=

25

38

= x

B = Select a brown candy.

B|B = Select a brown candy, given that a brown was already selected.

P(B)

P(B|B)

P(B and B)

= P(B) x P(B|B)

= 0.15

slide10

Example

The names of ten club members, four boys and six girls, are placed in a hat. Jack is one of the boys and Sally is one of the girls. Suppose that the first name drawn will be the president and the second will be the vice-president.

slide11

Example

Are the events independent or dependent?

dependent

slide12

130

Example

Find P(Jack, then a boy).

slide13

29

Example

Find P(a girl other than Sally, then a boy).

slide14

415

Example

Find P(a boy, then a girl).

slide15

415

Example

Find P(a girl, then a boy).

slide16

815

Example

What is the probability that one boy and one girl will be selected?

slide17

2

1

4

3

slide18

Probability of Independent Events

If A and B are independent events, then P(A and B) = P(A) x P(B).

slide19

2

1

1

4

4

5

Example 3

Find P(4 and tails).

slide20

26

13

12

=

=

=

16

13

12

=

=

Find P(4 and tails).

P(4)

P(T)

P(4andT)

= P(4) x P(T)

≈ 0.17

slide21

Example 4

A three-digit number is to be formed by drawing one of four slips of paper with the digits 1, 2, 3, and 4 from a hat. The first draw determines the first digit of the number to be formed, and so on.

slide22

Example 4

Digits can be used more than once, so the digit drawn is replaced in the hat before the next draw. What is the probability that the three-digit number formed is 123?

slide23

14

14

14

= x x

164

=

Find P(1 and 2 and 3).

P(1 and 2 and 3)

= P(1) x P(2) x P(3)

≈ 0.016

slide24

Example

The names of ten club members, four boys and six girls, are placed in a hat. Jack is one of the boys and Sally is one of the girls. Suppose names will be drawn to select a boy’s representative and a girl’s representative.

slide25

Example

Are the events independent or dependent?

independent

slide26

124

Example

What is the probability that Jack and Sally will be chosen as the representatives?

slide27

58

Example

What is the probability that neither Jack nor Sally will be chosen?

slide28

18

Example

What is the probability that Sally will be chosen but Jack will not?

slide29

Exercise

In a Christian high school of 250 students, 92 play only the piano, 12 play only the trumpet, and 8 play both.

slide30

Exercise

Use a Venn diagram to help you find the probability that each of the following will occur. Express your answer as both a fraction and a decimal rounded to the nearest thousandth.

slide31

225

= 0.08

Exercise

Find the probability that a student drawn at random plays the trumpet.

slide32

25

= 0.4

Exercise

Find the probability that a student drawn at random plays the piano.

slide33

4125

= 0.032

Exercise

Find the probability that a student drawn at random plays the piano and the trumpet.

slide34

25

= 0.4

Exercise

Find the probability that a student drawn at random plays the piano, given that he plays the trumpet.

slide35

Exercise

Does P(plays the piano and the trumpet) = P(plays the piano, given that he plays the trumpet)?

no