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A.P. Statistics - Conditional Probability 1. 4 coins are flipped. Find the following probabilities. Make a sample space.

a) that there are at least two heads. c) If there are at least 2 heads, that there are only 2 heads.

b) If there are at least 2 heads, there will be at least one more d) If there are at least 2 heads, that 2 heads never appear consecutively in the 4 flips.

2. Let a pair of fair dice be tossed. Find the following. Make a sample space.

a) the probability that at least one of the dice is a 4 b) the probability that the sum is 7 c) Given that the sum is 7, find the probability that one of the dice is 4 d) Given that at least one of the dice is 4, find the probability that the sum is 7

3. A cooler has 12 Coke's and 15 Pepsi's. 9 of the Coke's are diet Coke's and 5 of the Pepsi's are diet Pepsi's. A bottle is chosen at random. Find the following: Make a chart.

a) the probability that the bottle is a Coke. c) the probability that the bottle is a Diet Coke e) the probability that the bottle is a diet drink g) Given that the bottle is a diet drink, find the probability that the bottle is a Coke

b) the probability that the bottle is a Pepsi d) the probability that the bottle is a Diet Pepsi f) the probability that the bottle is not a diet drink h) Given that the bottle is diet drink, find the probability that the bottle is a Pepsi

i) Given that the bottle is a Coke, j) Given that the bottle is a Pepsi, find the probability that the bottle is a Diet Coke find the probability that the bottle is a Diet Pepsi

www.MasterMathMentor.com Stu Schwartz

4. Box A contains 8 light bulbs of which 3 are defective. Box B contains 12 light bulbs of which 5 are defective. A bulb is chosen at random from one of the 3 boxes. Make a chart.

a) Find the probability that the bulb is from Box A?b) Find the probability that the bulb is from box B? c) If the bulb chosen is from Box B, what is the probability that the bulb is defective?

d) What is the probability that the bulb is defective?

e) If the bulb is defective, what is the probability that it is from Box B?

5. A woman's club has 80% of its members married. 30% of the married women are pro-choice and 75% of the single women are pro-choice. If a woman is chosen at random, find the probability . Make a chart.

a) she is married c) she is pro-choice e) she is married and pro-choice g) Given that she is married, she is pro-choice i) Given she is single, she is pro-choice k) Given she is pro-choice, she is married

b) she is single d) she is pro-life f) she is single and pro-choice h) Given she is married, she is pro-life j) Given she is single, she is pro-life l) Given she is pro-life, she is single

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Stu Schwartz

CONDITIONAL PROBABILITY - Practice

Do these on a separate piece of paper and compare answers below. I have intentionally NOT reduced fractions so you can see where the answers come from. 1. Let a pair of fair dice be tossed. Find the following a) the probability that at least one of the dice is even b) the probability that the sum is 6 c) Given that the sum is 6, find the probability that one of the dice is even d) Given that at least one of the dice is even, find the probability that the sum is 6

e) Are the events (having at least one die even) and (having the sum = 6) independent? Why or why not? 2. 4 coins are flipped. Find the following: a) Find the probability that there are exactly two heads. c) If the first 3 coins are heads, find the probability that the fourth is also heads. b) if there are 2 heads, find the probability that they came consecutively. d) If the first two coins are the same, find the prob. that the last two coins are also the same.

3. A class has 10 girls and 5 boys. 7 of the girls are passing and 4 of the boys are passing. A student is chosen at random. Find the following probabilities a) that the student is a boy. b) that the student is a girl

c) that the student is a passing boy

d) that the student is a passing girl

e) that the student is passing

f) that the student is failing

g) Given that the student is passing, find the probability that the student is a boy

h) Given that the student is passing, find the probability that the student is a girl

i) Given that the student is a boy, j) Given that the student is a girl, find the probability that the student is passing find the probability that the student is passing

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4. A class has 100 students, 70 of which are boys. 70% of the boys are involved in sports, while 40% of the girls are involved in sports. A student is chosen at random. a) What is the probability that a student is a boy? b) What is the probability that the student is a girl? c) If the student chosen is a boy, what is the probability that he is involved in sports? d) What is the probability that the student is involved in sports? e) If the student chosen in involved in sports, what is the probability that he is a boy? f) What is the probability that the student is a boy or involved in sports? 5. In a certain college, 25% of the boys and 10% of the girls are on scholarships. The girls constitute 60% of the student body. If a student is chosen at random, find the probability a) that the person is a boy b) that the person is on a scholarship

c) Given that the person is a boy, d) Given that the person is on a scholarship, find the probability that he is on a scholarship. find the probability that he is a boy.

e) Given that the person is not on a scholarship find the probability that she is a girl.

f) that the person is a boy or on scholarship

6. Rooms I02, I04, and I06 contain only juniors and seniors in the following numbers. I02 I04 I06 Seniors 20 25 1 Juniors 5 3 26 I choose a room at random and then choose a student at random. Find the following probabilities: a. The student is from I06. c. If I chose someone from I04, it is a senior. e. If I chose a junior, he came from I06.

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b. The student is a senior. d. If I chose a senior, he came from I04. f. he is a senior or in I04

Stu Schwartz

Answers: 1) a.

27 36

b.

5 36

c.

2 2 d. e) Not independent - P(sum=6 |one die even)P(sum = 6) 5 27 1 2

2)

a.

6 3 b. 16 6

c.

d.

4 8

3)

a.

5 10 4 7 b. c. d. 15 15 15 15

e.

11 15

f.

4 4 7 4 7 g. h. i. j. 15 11 11 5 10

4) a. .70

b. .30

c.

47 70

d. .59 d.

.10 .16

5) a. .40 b. .16

1 3

c. .25

47 59 .54 e. .84

e.

f. 82 f.46

25 28

6) a.

! 1 $ ! 20$ ! 1 $ ! 25 $ ! 1$ ! 1 $ +# # +# # b. # # = .577 " 3% " 25% " 3 % " 28 % " 3% " 27% 25 P(I04 and Senior ) 80 = = .542 d. P(I04 | senior) = P(Senior) .577

c.

e. Since P(senior) = .577, P(junior) = .423. P(I06 | Junior) = f.

49 80

P(I06 and Junior ) P(Junior)

26 = 80 = .768 .423

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Stu Schwartz

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