Binomial distribution practice questions.

Which of the following situations can be modelled by a binomial distribution?

• The number of rolls of a die needed until the first six appears
• The number of red cards obtained when 5 cards are drawn from a deck without replacement
• The number of sixes obtained when a fair die is rolled 10 times
• The height of a randomly chosen student

Answer: The number of sixes obtained when a fair die is rolled 10 times

A binomial model needs a fixed number of independent trials with constant success probability; rolling a die 10 times and counting sixes satisfies all conditions.

X ~ B(10, 0.3). Which expression gives P(X = 2)?

• C(10,2) × 0.3² × 0.7⁸
• C(10,2) × 0.3⁸ × 0.7²
• 0.3² × 0.7⁸
• C(10,2) × 0.3² × 0.7²

Answer: C(10,2) × 0.3² × 0.7⁸

P(X = r) = C(n,r) pʳ (1−p)ⁿ⁻ʳ, so with n = 10, r = 2, p = 0.3 the failure exponent must be 10 − 2 = 8 and the binomial coefficient must be included.

X ~ B(8, 0.25). Calculate E(X).

• 1.5
• 2
• 6
• 2.5

Answer: 2

For a binomial distribution E(X) = np = 8 × 0.25 = 2; the value 1.5 is the variance np(1−p), a common mix-up.

X ~ B(n, 0.2) and E(X) = 6. Determine the value of n.

• 12
• 6
• 24
• 30

Answer: 30

E(X) = np gives 0.2n = 6, so n = 30.

X ~ B(20, 0.5). Determine the variance of X.

• 5
• 10
• 4
• 2.5

Answer: 5

Var(X) = np(1−p) = 20 × 0.5 × 0.5 = 5; the value 10 is E(X) = np, not the variance.

A fair coin is tossed 3 times. Determine the probability of obtaining at least one head.

• 1/8
• 1/2
• 7/8
• 3/8

Answer: 7/8

P(at least one head) = 1 − P(no heads) = 1 − (1/2)³ = 1 − 1/8 = 7/8; using the complement avoids adding three separate cases.