Independent & Dependent Events — Practice Set

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

Ask one question of every multi-stage problem: after the first event happens, has the pool changed? If yes the events are dependent and you must update both numbers; if no they are independent and you simply multiply.

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-IDE-001 · MCQ · 45s

A fair coin is flipped and a fair six-sided die is rolled. What is the probability of getting heads on the coin and a 4 on the die?

(A) 1/12
(B) 1/6
(C) 1/2
(D) 7/12
(E) 2/3

Q2 · STA-IDE-006 · Numeric Entry · 45s

The probability that Maya's train is late on any given day is 1/5, and the days are independent of one another. What is the probability that her train is late on both of the next two days? Enter your answer as a fraction in lowest terms.

Numeric entry — write the number.

Q3 · STA-IDE-007 · MCQ · 50s

Which of the following describes a pair of INDEPENDENT events?

(A) Drawing two cards from a standard deck without replacement
(B) Drawing a marble from a bag, keeping it, then drawing a second marble
(C) Rolling a fair die and flipping a fair coin
(D) Selecting two different students from one class, one at a time
(E) Dealing a card, setting it aside, and dealing a second card from the rest of the deck

Q4 · STA-IDE-009 · QC · 50s

A jar holds 5 red and 5 blue chips. One chip is drawn at random, its color is noted, and it is returned to the jar before a second chip is drawn.

Column A: P(the first chip is red) Column B: P(the second chip is red, given that the first chip was red)

Q5 · STA-IDE-010 · MCQ · 55s

A standard 52-card deck contains 4 aces. A card is drawn at random, returned to the deck, and the deck is shuffled; then a second card is drawn. What is the probability that both cards drawn are aces?

(A) 3/676
(B) 1/221
(C) 1/169
(D) 1/13
(E) 2/13

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · STA-IDE-011 · MCQ · 70s

A bag contains 5 red, 3 blue, and 2 green marbles. Two marbles are drawn at random one at a time, without replacement. What is the probability that the first is red and the second is blue?

(A) 1/9
(B) 3/20
(C) 1/6
(D) 1/3
(E) 1/2

Q7 · STA-IDE-016 · Numeric Entry · 75s

Two cards are drawn at random from a standard 52-card deck (26 red cards and 26 black cards), without replacement. What is the probability that both cards are red? Enter your answer as a fraction in lowest terms, in the form a/b.

Numeric entry — write the number.

Q8 · STA-IDE-018 · MCQ · 65s

A box holds 6 red and 4 blue balls. Two balls are drawn one at a time, without replacement. Given that the first ball drawn was blue, what is the probability that the second ball is red?

(A) 4/15
(B) 1/3
(C) 5/9
(D) 3/5
(E) 2/3

Q9 · STA-IDE-019 · Select all that apply · 90s

A bag contains 4 red, 3 blue, and 3 green marbles. Two marbles are drawn one at a time, without replacement. Select all of the following whose probability equals 2/15.

(A) P(first red, then blue)
(B) P(first blue, then red)
(C) P(first green, then green)
(D) P(first green, then red)
(E) P(first blue, then blue)

Q10 · STA-IDE-020 · MCQ · 80s

For two events A and B, P(A) = 0.5, P(B | A) = 0.4, and P(B | not A) = 0.4. Which statement about A and B is correct?

(A) They are independent, because P(B | A) = P(B | not A)
(B) They are independent, because P(A) = 0.5
(C) They are not independent, because P(B | A) is not equal to P(A)
(D) They are not independent, because P(A and B) = 0.2, which is not 0
(E) Independence cannot be decided without being told P(B)

Q11 · STA-IDE-021 · Numeric Entry · 80s

A bag holds 5 red and 5 blue marbles. Two marbles are drawn one at a time, without replacement. What is the probability that the two marbles are of different colors? Enter your answer as a fraction in lowest terms, in the form a/b.

Numeric entry — write the number.

Q12 · STA-IDE-022 · QC · 80s

One card is drawn at random from a standard 52-card deck, which has 26 red cards and 13 hearts, all hearts being red.

Column A: P(the card is red, given that it is a heart) Column B: P(the card is a heart, given that it is red)

Q13 · STA-IDE-025 · Select all that apply · 90s

Events A and B are mutually exclusive, and P(A) > 0 and P(B) > 0. Select all statements that must be true.

(A) P(A and B) = 0
(B) A and B are independent
(C) P(A or B) = P(A) + P(B)
(D) P(A | B) = 0
(E) P(B | A) = P(B)

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Level 3 · Hard — 12 questions · ~20 min

where 162+ is won or lost

Q14 · STA-IDE-027 · MCQ · 100s

A bag contains 4 red and 6 blue marbles. Three marbles are drawn at random, one at a time, without replacement. What is the probability that exactly two of them are red?

(A) 1/20
(B) 1/10
(C) 36/125
(D) 3/10
(E) 1/3

Q15 · STA-IDE-028 · QC · 95s

A jar holds 4 red marbles and 6 green marbles and no others. In each column below, two marbles are drawn at random, one at a time.

Column A: P(the second marble is red, given that the first marble is green), when the draws are made WITHOUT replacement Column B: P(the second marble is red), when the draws are made WITH replacement

Q16 · STA-IDE-029 · Numeric Entry · 95s

For two events A and B, P(A) = 0.6, P(B) = 0.5, and P(A or B) = 0.8. What is P(A | not B)? Enter your answer as a decimal.

Numeric entry — write the number.

Q17 · STA-IDE-030 · MCQ · 90s

A machine contains two components that fail independently of each other. Component A fails with probability 0.1 and component B fails with probability 0.2. The machine fails if at least one component fails. What is the probability that the machine does NOT fail?

(A) 0.28
(B) 0.70
(C) 0.72
(D) 0.90
(E) 0.98

Q18 · STA-IDE-031 · Select all that apply · 110s

Two cards are drawn at random from a standard 52-card deck (13 cards in each of the four suits), without replacement. Select all statements that are true.

(A) P(both cards are spades) = 1/17
(B) P(the first card is a spade and the second is a heart) = 1/16
(C) P(at least one card is a spade) = 1 - (39/52)(38/51)
(D) P(the second card is a spade, given that the first card is a spade) = 4/17
(E) The two draws are independent events

Q19 · STA-IDE-032 · MCQ · 95s

A bag holds 5 red and 3 blue balls. Balls are drawn at random one at a time, without replacement, until a blue ball appears. What is the probability that the first blue ball appears on exactly the second draw?

(A) 9/56
(B) 15/64
(C) 15/56
(D) 3/8
(E) 3/7

Q20 · STA-IDE-033 · QC · 90s

Events A and B are independent, with P(A) = p and P(B) = q, where 0 < p < 1 and 0 < q < 1.

Column A: P(A or B) Column B: p + q

Q21 · STA-IDE-034 · Numeric Entry · 100s

A box contains 4 red, 3 blue, and 2 green balls. Two balls are drawn at random, without replacement. Given that the two balls drawn are the same color, what is the probability that both are red? Enter your answer as a fraction in lowest terms, in the form a/b.

Numeric entry — write the number.

Q22 · STA-IDE-035 · MCQ · 95s

At a school, 70% of students play a sport and 40% play a musical instrument. Playing a sport and playing an instrument are independent. If one student is chosen at random, what is the probability that the student does exactly one of the two activities?

(A) 0.28
(B) 0.30
(C) 0.46
(D) 0.54
(E) 0.82

Q23 · STA-IDE-036 · Select all that apply · 100s

A fair six-sided die is rolled three times. Select all of the following probabilities that equal 1/216.

(A) P(a 6 on all three rolls)
(B) P(the rolls are 1, then 2, then 3, in that order)
(C) P(the three rolls are all different numbers, in any order)
(D) P(a 6 on the first two rolls and a non-6 on the third)
(E) P(the rolls are 4, then 4, then 4)

Q24 · STA-IDE-041 · QC · 105s

A box holds 3 red and 3 blue balls. Two balls are drawn at random, one at a time, without replacement.

Column A: P(the second ball is red, given that the first ball is red) Column B: P(the second ball is red)

Q25 · STA-IDE-042 · Numeric Entry · 105s

Two fair six-sided dice, one red and one white, are rolled. Given that at least one of the dice shows a 3, what is the probability that both dice show a 3? Enter your answer as a fraction in lowest terms, in the form a/b.

Numeric entry — write the number.

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Level 4 · Extreme — 5 questions · ~10 min

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

Q26 · STA-IDE-037 · MCQ · 120s

A bag contains 3 red and 5 blue marbles and no others. Two marbles are drawn at random, one at a time, without replacement. Given that the SECOND marble drawn is red, what is the probability that the first marble drawn was also red?

(A) 3/28
(B) 1/4
(C) 2/7
(D) 3/8
(E) 3/7

Q27 · STA-IDE-038 · QC · 120s

A bag contains n red and n blue marbles, where n is an integer and n >= 2. Two marbles are drawn at random, without replacement.

Column A: P(the two marbles are the same color) Column B: P(the two marbles are different colors)

Q28 · STA-IDE-039 · Numeric Entry · 130s

A bag contains 4 white and 6 black balls. Balls are drawn at random one at a time, without replacement. What is the probability that the fourth ball drawn is the second white ball? Enter your answer as a fraction in lowest terms, in the form a/b.

Numeric entry — write the number.

Q29 · STA-IDE-040 · Select all that apply · 125s

For two events A and B, P(A) = 1/3, P(B) = 1/2, and P(A | B) = 1/4. Select all statements that are true.

(A) P(A and B) = 1/8
(B) A and B are independent
(C) P(B | A) = 3/8
(D) P(A or B) = 5/6
(E) P(neither A nor B) = 7/24

Q30 · STA-IDE-043 · MCQ · 130s

A fair six-sided die is rolled twice. Exactly one of the following pairs of events is independent. Which one?

(A) 'The first roll is even' and 'the sum of the two rolls is 8'
(B) 'The first roll is even' and 'the sum of the two rolls is 7'
(C) 'The first roll is a 6' and 'the product of the two rolls is even'
(D) 'Both rolls are even' and 'the sum of the two rolls is even'
(E) 'The sum of the two rolls is 7' and 'the sum of the two rolls is 8'

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