Probability: Rules, Conditional, Independence — HS Statistics practice questions

What is the expected value of one roll of a fair six-sided die?

1. 3
2. 6
3. 3.5
4. 21

Answer: C — 3.5

C) Correct — (1 + 2 + 3 + 4 + 5 + 6)/6 = 21/6. A) Rounds down; the expected value need not be a possible roll. B) Is the largest outcome. D) Adds the outcomes without dividing by 6.

How many 3-letter arrangements can be made from 5 different letters, with no repeats?

1. 10
2. 125
3. 60
4. 15

Answer: C — 60

C) Correct — Order matters: 5 · 4 · 3. A) Counts combinations, ignoring order. B) Allows letters to repeat. D) Multiplies 5 × 3.

P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.2. Find P(A or B).

1. 0.9
2. 0.2
3. 0.7
4. 0.5

Answer: C — 0.7

C) Correct — P(A) + P(B) − P(A and B) = 0.5 + 0.4 − 0.2. A) Forgets to subtract the overlap, counting it twice. B) Is P(A and B). D) Uses only P(A).

A fair coin lands heads 5 times in a row. What is P(heads) on the next flip?

1. Less than 1/2, because tails is due
2. More than 1/2, because heads is on a streak
3. 1/2
4. 1/64

Answer: C — 1/2

C) Correct — The flips are independent; the coin has no memory. A) The gambler's fallacy: past flips do not affect the next one. B) Streaks do not change a fair coin's probability. D) Is the probability of 6 heads in a row, not of the next flip.

Two dice are rolled. How many outcomes are in the sample space?

1. 12
2. 11
3. 36
4. 6

Answer: C — 36

C) Correct — 6 outcomes for the first die × 6 for the second. A) Adds 6 + 6. B) Counts the possible sums, 2 through 12. D) Counts one die.

Of 200 students, 80 play a sport, and 30 of those are seniors. What is P(senior | plays a sport)?

1. 0.15
2. 0.40
3. 0.375
4. 0.625

Answer: C — 0.375

C) Correct — 30 of the 80 athletes: 30/80. A) Is P(senior and plays a sport) = 30/200. B) Is P(plays a sport) = 80/200. D) Is the share of athletes who are not seniors.

A fair coin is flipped 3 times. What is P(at least one head)?

1. 3/8
2. 7/8
3. 1/2
4. 3/2

Answer: B — 7/8

B) Correct — 1 − P(no heads) = 1 − 1/8. A) Is P(exactly one head). C) Uses the probability for one flip. D) Adds 1/2 three times; probabilities cannot exceed 1.

P(rain) = 0.35. What is P(no rain)?

1. 0.35
2. 0.65
3. 0.5
4. 1.35

Answer: B — 0.65

B) Correct — Complement: 1 − 0.35. A) Copies the probability of rain. C) Assumes rain and no rain are equally likely. D) Adds 1 instead of subtracting from 1.

Events A and B are independent exactly when:

1. P(A and B) = P(A)·P(B)
2. P(A and B) = 0
3. P(A) = P(B)
4. P(A) + P(B) = 1

Answer: A — P(A and B) = P(A)·P(B)

A) Correct — Knowing one happened does not change the chance of the other. B) Defines mutually exclusive events. C) Equal probabilities say nothing about independence. D) Describes complements in some cases.

Mutually exclusive (disjoint) events?

Can't both happen. P(A or B) = P(A) + P(B).

Sample space?

Set of all possible outcomes of a random experiment. P(sample space) = 1.

Bayes' theorem?

P(A|B) = P(B|A) × P(A) / P(B). Updates probability with new info.

Complement rule?

P(not A) = 1 - P(A).