Which of the following random variables follow a binomial distribution?
(A) The number of heads obtained when a fair coin is tossed 10 times
(B) The number of red balls obtained when 3 balls are drawn one by one without replacement from a bag containing 5 red and 5 black balls
(C) The number of sixes obtained when a fair die is thrown 6 times
(D) The number of tosses of a fair coin needed to obtain the first head
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
A random variable is binomial when it counts the successes in a fixed number of trials that are independent, each trial having only two outcomes and the same probability of success.
So each variable must count successes in a fixed number of independent trials with an unchanging p.
Checking the statements
(A) Heads in 10 tosses — Correct. n = 10 independent tosses, each with \(p = \frac{1}{2}\).
(B) Red balls in 3 draws without replacement — Incorrect. Without replacement the chance of red changes after each draw, so the trials are not independent.
(C) Sixes in 6 throws — Correct. n = 6 independent throws, each with \(p = \frac{1}{6}\).
(D) Tosses needed for the first head — Incorrect. The number of trials is not fixed; it is the variable itself.