Question

Consider a sequence of independent Bernoulli trials with probability of success in each trial being $\dfrac{1}{3}$. Let $X$ denote the number of trials required to get the second success. Then $P(X \ge 5)$ equals
 

• A. $\dfrac{3}{7}$
• B. $\dfrac{16}{27}$
• C. $\dfrac{16}{21}$
• D. $\dfrac{9}{13}$

Hint:

For "number of trials until $r$ successes," use the Negative Binomial distribution and cumulative subtraction for probabilities like $P(X \ge k)$.

Answer 1

The Correct Option is B

Step 1: Distribution identification.
$X$ follows a Negative Binomial distribution (number of trials to get 2 successes).

Step 2: Formula.
\[ P(X = k) = \binom{k - 1}{1} p^2 (1 - p)^{k - 2}, p = \frac{1}{3}. \] Then \[ P(X \ge 5) = 1 - P(X \le 4) = 1 - [P(2) + P(3) + P(4)]. \]

Step 3: Compute each term.
\[ P(2) = \binom{1}{1} \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^0 = \frac{1}{9}, P(3) = \binom{2}{1} \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^1 = \frac{4}{27}, \] \[ P(4) = \binom{3}{1} \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^2 = \frac{8}{27}. \] \[ P(X \le 4) = \frac{1}{9} + \frac{4}{27} + \frac{8}{27} = \frac{11}{27}. \] Thus, \[ P(X \ge 5) = 1 - \frac{11}{27} = \frac{16}{27}. \]

Step 4: Conclusion.
\[ \boxed{P(X \ge 5) = \frac{16}{27}}. \]