Compound Probability — Practice Set

Bank code: STA-CPR · Section: statistics · 30 questions — Easy 5 · Medium 8 · Hard 12 · Extreme 5

AND multiplies, OR adds minus the overlap, and without replacement both numbers shrink.

Every question below is live in the question bank under the ID shown — the sheet and the portal are the same questions. Attempt a level with the clock running, then check Part C.

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Part A — Questions

Quantitative Comparison — the four choices are always the same, so they are not reprinted: (A) Column A is greater · (B) Column B is greater · (C) The two quantities are equal · (D) The relationship cannot be determined from the information given

Level 1 · Easy — 5 questions · ~4 min

warm-up — these must be automatic

Q1 · STA-CPR-001 · MCQ · 40s

A fair coin is flipped twice. What is the probability that both flips come up heads?

(A) 1/4
(B) 1/3
(C) 1/2
(D) 3/4
(E) 1

Q2 · STA-CPR-007 · MCQ · 55s

The probability that it rains on Saturday is 0.4 and the probability that it rains on Sunday is 0.3. The two days are independent. What is the probability that it rains on at least one of the two days?

(A) 0.12
(B) 0.42
(C) 0.58
(D) 0.70
(E) 0.88

Q3 · STA-CPR-010 · MCQ · 50s

A fair six-sided die is rolled three times. What is the probability that every one of the three rolls shows a number less than 3?

(A) 1/216
(B) 1/27
(C) 1/9
(D) 1/8
(E) 1/3

Q4 · STA-CPR-011 · QC · 45s

A fair six-sided die is rolled twice.

Column A: P(both rolls show an even number) Column B: P(both rolls show an odd number)

Q5 · STA-CPR-013 · Numeric Entry · 55s

A box contains 6 white balls and 4 black balls and no others. Two balls are drawn at random, one at a time, WITHOUT replacement. What is the probability that both balls are white? Enter your answer as a fraction in lowest terms.

Numeric entry — write the number.

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Level 2 · Medium — 8 questions · ~11 min

two or three steps, one planted trap each

Q6 · STA-CPR-015 · MCQ · 70s

Two fair six-sided dice are rolled. What is the probability that the sum of the two numbers rolled is 7?

(A) 1/12
(B) 1/11
(C) 1/7
(D) 1/6
(E) 7/36

Q7 · STA-CPR-017 · MCQ · 70s

At a company, 60% of the employees are female and 40% are male. Among the female employees, 30% are managers. Among the male employees, 20% are managers. What is the probability that an employee chosen at random is female AND a manager?

(A) 0.08
(B) 0.18
(C) 0.26
(D) 0.30
(E) 0.90

Q8 · STA-CPR-020 · Select all that apply · 90s

A fair six-sided die is rolled twice. Which of the following statements are true? Select all that apply.

(A) P(both rolls show a 1) = 1/36
(B) P(the first roll is even and the second roll is odd) = 1/4
(C) P(at least one roll shows a 6) = 1/3
(D) P(the two rolls show the same number) = 1/36
(E) P(the second roll is larger than the first) = 1/2

Q9 · STA-CPR-022 · Numeric Entry · 85s

A box contains 5 defective items and 15 non-defective items. Two items are selected at random WITHOUT replacement. What is the probability that at least one of the two selected items is defective? Enter your answer as a fraction in lowest terms.

Numeric entry — write the number.

Q10 · STA-CPR-026 · QC · 85s

A bag contains 3 red balls and 3 blue balls and no other balls. Two balls are drawn at random WITHOUT replacement.

Column A: P(both balls are red) Column B: P(one ball is red and the other is blue)

Q11 · STA-CPR-027 · Select all that apply · 90s

Events A and B are independent, with P(A) = 0.6 and P(B) = 0.5. Which of the following statements are true? Select all that apply.

(A) P(A and B) = 0.30
(B) P(A or B) = 1.10
(C) P(A | B) = 0.6
(D) P(neither A nor B occurs) = 0.20
(E) P(A and not B) = 0.24

Q12 · STA-CPR-029 · MCQ · 85s

Box A contains 3 red marbles and 2 white marbles. Box B contains 2 red marbles and 6 white marbles. A fair coin is flipped: if it lands heads, one marble is drawn at random from Box A; if it lands tails, one marble is drawn at random from Box B. What is the probability that the marble drawn is red?

(A) 3/20
(B) 5/13
(C) 17/40
(D) 23/40
(E) 17/20

Q13 · STA-CPR-031 · MCQ · 85s

A machine produces items independently, and each item is defective with probability 0.1. Three items are produced. What is the probability that exactly one of the three is defective?

(A) 0.001
(B) 0.081
(C) 0.243
(D) 0.27
(E) 0.729

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Level 3 · Hard — 12 questions · ~20 min

where 162+ is won or lost

Q14 · STA-CPR-034 · MCQ · 105s

An urn contains 4 red, 3 green, and 5 blue balls. Three balls are drawn at random WITHOUT replacement. What is the probability that the three balls are all of different colors?

(A) 5/144
(B) 1/22
(C) 3/22
(D) 5/24
(E) 3/11

Q15 · STA-CPR-035 · Numeric Entry · 100s

A bag contains 3 red chips and 5 blue chips and no others. Chips are drawn one at a time at random WITHOUT replacement. What is the probability that the SECOND chip drawn is red? Enter your answer as a fraction in lowest terms.

Numeric entry — write the number.

Q16 · STA-CPR-036 · MCQ · 100s

Two fair six-sided dice, one red and one white, are rolled. Given that the sum of the two numbers rolled is 8, what is the probability that at least one of the dice shows a 6?

(A) 1/18
(B) 1/5
(C) 11/36
(D) 1/3
(E) 2/5

Q17 · STA-CPR-037 · MCQ · 105s

Urn X contains 2 red and 3 white balls. Urn Y contains 4 red and 1 white ball. One ball is drawn at random from Urn X and placed into Urn Y. A ball is then drawn at random from Urn Y. What is the probability that this second ball is red?

(A) 2/3
(B) 11/15
(C) 3/4
(D) 4/5
(E) 5/6

Q18 · STA-CPR-038 · QC · 105s

A bag contains 4 red marbles and 6 blue marbles and no others. Two marbles are drawn at random.

Column A: P(the two marbles are different colors) when the two draws are made WITH replacement Column B: P(the two marbles are different colors) when the two draws are made WITHOUT replacement

Q19 · STA-CPR-039 · Select all that apply · 100s

A fair coin is flipped 4 times. Which of the following statements are true? Select all that apply.

(A) P(exactly 0 heads) = 1/16
(B) P(at least 3 heads) = 5/16
(C) P(exactly 2 heads) = 1/4
(D) P(the first two flips are both heads) = 1/4
(E) P(all four flips show the same face) = 1/16

Q20 · STA-CPR-040 · Numeric Entry · 100s

A drawer contains 4 black socks and 6 white socks and no others. Three socks are drawn at random WITHOUT replacement. What is the probability that the three socks are NOT all the same color? Enter your answer as a fraction in lowest terms.

Numeric entry — write the number.

Q21 · STA-CPR-041 · QC · 100s

A bag contains 2 red marbles and 6 blue marbles and no others. Three marbles are drawn at random WITHOUT replacement.

Column A: P(at least one of the three marbles is red) Column B: 2/3

Q22 · STA-CPR-042 · MCQ · 110s

A test for a certain disease is 90% accurate in both directions: it reports positive for 90% of the people who have the disease, and negative for 90% of the people who do not. In a population in which 1% of people have the disease, one person is chosen at random and tests positive. What is the probability that this person actually has the disease?

(A) 0.9%
(B) 8 1/3%
(C) 9 1/11%
(D) 10%
(E) 90%

Q23 · STA-CPR-043 · Numeric Entry · 100s

Two cards are drawn one at a time at random WITHOUT replacement from a standard 52-card deck. What is the probability that the two cards are both aces or both face cards (jack, queen, or king)? Enter your answer as a fraction in lowest terms.

Numeric entry — write the number.

Q24 · STA-CPR-044 · Select all that apply · 105s

A jar contains 2 red marbles and 4 white marbles and no others. Three marbles are drawn at random, one at a time, WITHOUT replacement. Which of the following statements are true? Select all that apply.

(A) P(all three marbles are white) = 1/5
(B) P(at least one of the three marbles is red) = 4/5
(C) P(exactly one of the three marbles is red) = 4/9
(D) P(the first marble is red and the other two are white) = 1/5
(E) P(the second marble drawn is white) = 3/5

Q25 · STA-CPR-045 · QC · 100s

A coin, not necessarily fair, comes up heads with probability p, where 0 < p < 1. The coin is flipped twice and the flips are independent.

Column A: P(at least one head in the two flips) Column B: 2p - p^2

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Level 4 · Extreme — 5 questions · ~10 min

165+ — expect to need the insight, not the grind

Q26 · STA-CPR-046 · Numeric Entry · 125s

A fair coin is flipped 6 times. What is the probability that the number of heads is greater than the number of tails? Enter your answer as a fraction in lowest terms.

Numeric entry — write the number.

Q27 · STA-CPR-047 · MCQ · 120s

An urn contains 3 red, 4 blue, and 5 green balls. Balls are drawn one at a time at random WITHOUT replacement until a red ball is drawn. What is the probability that exactly 3 balls are drawn in total, that is, that the third ball drawn is the first red one?

(A) 3/22
(B) 9/64
(C) 9/55
(D) 1/4
(E) 6/11

Q28 · STA-CPR-048 · QC · 120s

A biased coin comes up heads with probability 2/3 on every flip, and the flips are independent. The coin is flipped repeatedly until heads appears.

Column A: P(heads first appears on the 3rd flip) Column B: 3 x P(heads first appears on the 4th flip)

Q29 · STA-CPR-049 · Numeric Entry · 125s

Four people are chosen at random. Each person's birthday is equally likely to fall on any of the 7 days of the week, independently of the other three. What is the probability that at least two of the four people were born on the same day of the week? Enter your answer as a fraction in lowest terms.

Numeric entry — write the number.

Q30 · STA-CPR-050 · Select all that apply · 130s

Events A, B, and C are mutually independent, with P(A) = 1/2, P(B) = 1/3, and P(C) = 1/4. Which of the following statements are true? Select all that apply.

(A) P(A and B and C) = 1/24
(B) P(none of A, B, C occurs) = 1/4
(C) P(exactly one of A, B, C occurs) = 1/4
(D) P(at least one of A, B, C occurs) = 3/4
(E) P(at least two of A, B, C occur) = 7/24

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