Question

Find the probability that a person tossing 3 fair coins get either all heads or all tails for the second time on the fifth trial.

Answer 1

Answer on Question #44956 – Math - Statistics and Probability

Find the probability that a person tossing 3 fair coins get either all heads or all tails for the second time on the fifth trial.

Solution

Let $X$ be the number of trials (tossing) until get either all heads or all tails for the second time. Then $X$ is a negative binomial random variable,

$$b(x; k, p) = \binom{x - 1}{k - 1} p^k q^{x - k}, x = k, k + 1, k + 2, \ldots, k = 2.$$

With,

$$p = P(\text{success}) = P(\text{all heads or all tails}) = \frac{2}{8} = \frac{1}{4}.$$ $$q = 1 - p = 1 - \frac{1}{4} = \frac{3}{4}.$$ $$P[X = 5] = b\left(5; 2, \frac{1}{4}\right) = \binom{4}{1} \left(\frac{1}{4}\right)^2 \left(\frac{3}{4}\right)^3 = \frac{27}{256}.$$

Answer: $\frac{27}{256}$.

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