Binomial Distribution

Question

A biased coin showing heads with probability 1/3 is tossed five independent times in succession; what is the probability of obtaining exactly two heads among these five tosses?

RankGuru · Accepted Answer

The binomial distribution governs the number of successes in a fixed number of independent trials each with the same success probability, giving P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ. This is the canonical JEE Advanced repeated-Bernoulli-trials model. Here n = 5, p = 1/3 for heads, 1 − p = 2/3, and we need exactly r = 2 heads. Thus P(X = 2) = C(5, 2)(1/3)²(2/3)³ = 10 × (1/9) × (8/27) = 10 × 8 / 243 = 80/243. Option 40/243 drops the combinatorial factor by using C(5,2) = 5 instead of 10. Option 10/243 keeps only the binomial coefficient times a single 1/27 and omits the powers of 2/3 correctly. Option 16/81 ignores the coefficient and mismatches the exponents. The result follows directly from the binomial probability formula, with C(5,2) = 10 counting the arrangements of two heads among five tosses. Plausibility check: 80/243 ≈ 0.329 is a valid probability, and since the expected number of heads is np = 5/3 ≈ 1.67, having exactly two heads is plausibly one of the most likely outcomes, consistent with this moderate value.

Binomial Distribution

Select the correct option:

🔒 Solution Hidden from View

More statistics and probability Practice Questions

A binomial random variable representing successes in n trials has mean equal to 4 and variance equal...

In a binomial setting where a marksman hits a target with probability 1/4 on each independent shot, ...

In a binomial distribution, the number of trials is denoted by

When every observation in a dataset is first multiplied by three and then increased by five, how doe...

Suppose events A and B satisfy P(A) = 1/2, P(B) = 1/3, and the conditional probability P(A given B) ...

Two distinct groups contain twenty and thirty students with mean marks of sixty and seventy respecti...

A group of ten numbers has a recorded mean of fifteen, but later one entry originally read as eight ...

From a committee containing seven men and five women, a subcommittee of three persons is selected at...

A fair six-faced die is rolled once and the random variable X equals the number shown on the top fac...

Two friends agree to meet between five and six o'clock, each arriving at a uniformly random time and...

Two players alternately roll a fair die starting with the first player, and whoever first rolls a si...

A diagnostic test detects a disease present in one percent of a population with sensitivity ninety-n...

About This Question

Solution