An unbiased coin is tossed n times. The probability of getting at least one tail is p and the probability of at least two tails is q and p - q = $$\frac{5}{32}$$. What is the value of n?

Correct option is B

​$$\begin{aligned}&\text{The probability of getting at least one tail is:} \\&\qquad p = 1 - \left(\frac{1}{2}\right)^n \\&\text{This is because the only way to avoid getting at least one tail is if all tosses result in heads, which occurs with} \\&\text{probability } \left(\frac{1}{2}\right)^n. \\\\&\text{ the probability of getting at least two tails is:} \\&\qquad q = 1 - \left[ \left(\frac{1}{2}\right)^n + n \times \left(\frac{1}{2}\right)^n \right] \\&\text{Simplifying this expression:} \\&\qquad q = 1 - (n + 1) \left(\frac{1}{2}\right)^n \\\\&\text{We are given that:} \\&\qquad p - q = \frac{5}{32}\end{aligned}$$​​

​$$\begin{aligned}&\left(1 - \left(\frac{1}{2}\right)^n\right) - \left(1 - (n + 1) \left(\frac{1}{2}\right)^n\right) = \frac{5}{32} \\&\text{Simplifying this equation:} \\&\qquad (n + 1) \left(\frac{1}{2}\right)^n - \left(\frac{1}{2}\right)^n = \frac{5}{32} \\&\qquad n \left(\frac{1}{2}\right)^n = \frac{5}{32} \\&\text{Now, we check different values of } n. \\&\qquad \bullet \ \text{For } n = 5: \\&\qquad \qquad 5 \left(\frac{1}{2}\right)^5 = 5 \times \frac{1}{32} = \frac{5}{32}\end{aligned}$$​​