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

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

Independent EventsIndependent Events

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

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

Dependent EventsDependent Events

Dick

Ellen

Greg

Carl

Heather

Alan

Probability of Dependent Events

Probability of Dependent Events

If A and B are dependent events, then

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

If A and B are dependent events, then

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

Probability of Dependent Events

Probability of Dependent Events

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

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

Example 1Example 1Select 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.

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.

B = Select a boy’s name.B = Select a boy’s name.

P(B)P(B)

≈ 0.27≈ 0.27

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

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

4646

== 2323

== P(G|B)P(G|B) 2525

==

P(B and G)P(B and G) = P(B) x P(G|B)= P(B) x P(G|B)2323

= x= x 2525

4154

15==

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?

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?

Example 2Example 2

B = Select a brown candy.B = Select a brown candy.

P(B)P(B)

= 0.15= 0.15

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

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

10251025

== 2525

== P(B|B)P(B|B) 9249

24== 3

838

==

P(B and B)P(B and B) = P(B) x P(B|B)= P(B) x P(B|B)2525

= x= x 3838

3203

20==

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.

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.

ExampleExample

Are the events independent or dependent?Are the events independent or dependent?

dependentdependent

ExampleExample

Find P(Jack, then a boy).Find P(Jack, then a boy).

1301

30

ExampleExample

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

2929

ExampleExample

Find P(a boy, then a girl).Find P(a boy, then a girl).

4154

15

ExampleExample

Find P(a girl, then a boy).Find P(a girl, then a boy).

4154

15

ExampleExample

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

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

8158

15

ExampleExample

11

44 33

22

Probability of Independent Events

Probability of Independent Events

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

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

Find P(4 and tails).Find P(4 and tails).

11

4455

44

1122

Example 3Example 3

P(4)P(4)

≈ 0.17≈ 0.17

2626

== 1313

== P(T)P(T) 1212

==

P(4 and T)P(4 and T) = P(4) x P(T)= P(4) x P(T)

1616

==

Find P(4 and tails).Find P(4 and tails).

1313

== 1212

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.

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.

Example 4Example 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?

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?

Example 4Example 4

≈ 0.016≈ 0.016

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

1641

64==

Find P(1 and 2 and 3).Find P(1 and 2 and 3).

1414

= x x= x x1414

1414

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.

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.

ExampleExample

Are the events independent or dependent?Are the events independent or dependent?

independentindependent

ExampleExample

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

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

1241

24

ExampleExample

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

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

5858

ExampleExample

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

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

1818

ExampleExample

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

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

ExerciseExercise

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.

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.

ExerciseExercise

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

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

2252

25= 0.08= 0.08

ExerciseExercise

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

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

2525

= 0.4= 0.4

ExerciseExercise

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

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

41254

125= 0.032= 0.032

ExerciseExercise

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

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

2525

= 0.4= 0.4

ExerciseExercise

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

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

nono

ExerciseExercise