Probability and Conditional Probability Flashcards

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble is drawn at random, what is the probability that it is NOT green?

Answer: 4/5

There are 8 non-green marbles out of 10 total, so P(not green) = 8/10 = 4/5.

A survey of 200 students found that 120 play sports and 80 play an instrument. If 40 students do both, how many students do neither?

Answer: 40

Using inclusion-exclusion: sports or instrument = 120 + 80 − 40 = 160, so neither = 200 − 160 = 40.

The two-way table shows pet ownership among 150 households. Cat No Cat Dog 30 45 No Dog 25 50 Given that a household has a dog, what is the probability it also has a cat?

Answer: 2/5

There are 30 + 45 = 75 dog households; 30 have both, so P(cat | dog) = 30/75 = 2/5.

A standard die is rolled twice. What is the probability of getting a 4 on the first roll and a number greater than 2 on the second roll?

Answer: 1/9

P(4) = 1/6 and P(>2) = 4/6 = 2/3; since independent, multiply: (1/6)(2/3) = 2/18 = 1/9.

In a class of 30 students, 18 passed the math test and 12 passed the science test. If 6 students passed both, what is the probability that a randomly selected student passed at least one test?

Answer: 4/5

Students passing at least one = 18 + 12 − 6 = 24; P = 24/30 = 4/5.

A jar has 4 red and 6 blue chips. Two chips are drawn without replacement. What is the probability that both chips are red?

Answer: 2/15

P(1st red) = 4/10; P(2nd red | 1st red) = 3/9; product = 12/90 = 2/15.

The table below shows 300 survey responses by age group and preference. Prefer A Prefer B Under 30 90 60 30 and over 80 70 What is the probability that a randomly chosen respondent under 30 prefers A?

Answer: 3/5

There are 90 + 60 = 150 respondents under 30; 90 prefer A, so P = 90/150 = 3/5.

A two-way table records 400 employees by department and satisfaction level. Satisfied Neutral Dissatisfied Engineering 80 40 30 Marketing 60 60 50 Operations 40 20 20 What is the probability a randomly selected employee is from Marketing and Dissatisfied?

Answer: 1/8

Marketing & Dissatisfied = 50 out of 400 total; P = 50/400 = 1/8.

P(A) = 0.6 and P(A and B) = 0.18. If A and B are independent, what is P(B)?

Answer: 0.30

Since A and B are independent, P(B) = P(A and B)/P(A) = 0.18/0.6 = 0.30.

A box has 3 red, 5 white, and 2 black balls. One ball is drawn and not replaced; then a second is drawn. What is the probability the second ball is red, given the first was white?

Answer: 1/3

After removing a white ball, 9 balls remain: 3 red; P = 3/9 = 1/3.

A game wheel has 4 red, 3 blue, and 1 green section. A player spins twice. What is the probability of getting green both times?

Answer: 1/64

P(green) = 1/8; for two independent spins, P = (1/8)² = 1/64.

In a class of 25 students, the probability that a randomly chosen student plays guitar is 0.32. What is the expected number of guitar players?

Answer: 8

Expected value = 25 × 0.32 = 8.

The two-way table shows 500 people by exercise frequency and BMI category. Normal BMI High BMI Exercise daily 180 70 Rarely exercise 90 160 What fraction of people with a Normal BMI exercise daily?

Answer: 2/3

Normal BMI total = 180 + 90 = 270; exercise daily = 180; P = 180/270 = 2/3.

A bag has 7 green and 3 yellow tokens. Two tokens are drawn with replacement. What is the probability of drawing one of each color?

Answer: 42/100

P(green then yellow) + P(yellow then green) = (7/10)(3/10) + (3/10)(7/10) = 42/100 = 21/50.

A class has 24 students. The probability of any student being absent on a given day is 0.05. What is the expected number of absent students?

Answer: 1.2

E = n × p = 24 × 0.05 = 1.2.

The two-way table shows 200 voters by party and stance on a bill. Favor Oppose Democrat 70 30 Republican 20 80 Given that a voter favors the bill, what is the probability the voter is a Republican?

Answer: 2/9

Total favoring = 70 + 20 = 90; Republicans favoring = 20; P = 20/90 = 2/9.

Cards numbered 1 through 20 are shuffled. One card is drawn at random. What is the probability the card shows a multiple of 3 or a multiple of 5?

Answer: 2/5

Multiples of 3: {3,6,9,12,15,18} = 6; multiples of 5: {5,10,15,20} = 4; overlap {15} = 1; union = 9; P = 9/20.

A bag has 5 red and 5 blue chips. Two chips are drawn without replacement. What is the probability both are the same color?

Answer: 4/9

P(RR) = (5/10)(4/9) = 20/90; P(BB) = (5/10)(4/9) = 20/90; total = 40/90 = 4/9.

A company's hiring data shows: 40% of applicants pass the written test and, of those, 30% pass the interview. What percent of all applicants get hired?

Answer: 12%

P(pass written and interview) = 0.40 × 0.30 = 0.12 = 12%.

The two-way table shows 300 students by study method and test result. Pass Fail Study group 110 40 Study alone 90 60 What is the probability a student who failed studied alone?

Answer: 3/5

Total failing = 40 + 60 = 100; studied alone and failed = 60; P = 60/100 = 3/5.