Question

Consider a sequence of independent Bernoulli trials with probability of success in each trial being $\dfrac{1}{5}$. Then which of the following statements is/are TRUE?
 

• A. Expected number of trials required to get the first success is 5
• B. Expected number of successes in first 50 trials is 10
• C. Expected number of failures preceding the first success is 4
• D. Expected number of trials required to get the second success is 10

Hint:

For Bernoulli trials, geometric and negative binomial expectations are reciprocal to the success probability: $E(X) = \frac{r}{p}$.

Answer 1

The Correct Option is A, B, C, D

Step 1: Define the model.
Let $p = \dfrac{1}{5}$ be the probability of success. We use properties of geometric and negative binomial distributions.

Step 2: Expected number of trials for the first success.
For a geometric distribution, \[ E(X) = \frac{1}{p} = 5. \] Hence, (A) is true.

Step 3: Expected number of successes in 50 trials.
For a binomial distribution $B(n=50, p=\frac{1}{5})$, \[ E(X) = np = 50 \times \frac{1}{5} = 10. \] Hence, (B) is true.

Step 4: Expected number of failures before the first success.
For a geometric distribution, the expected number of failures is \[ E(X - 1) = \frac{1 - p}{p} = 4. \] Hence, (C) is true.

Step 5: Expected number of trials for the second success.
For the negative binomial distribution (r = 2, p = 1/5): \[ E(X) = \frac{r}{p} = \frac{2}{1/5} = 10. \] Hence, (D) is also true.

Step 6: Conclusion.
\[ \boxed{\text{All statements (A), (B), (C), and (D) are true.}} \]