Question

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is :

• A. $\frac{1}{32}$
• B. $\frac{3}{16}$
• C. $\frac{5}{16}$
• D. $\frac{1}{2}$

Hint:

For any binomial distribution where $p = q = 1/2$, the probability of getting an odd number of successes is always exactly $0.5$.

Answer 1

The Correct Option is D

Step 1: $P(Odd) = 1/2, P(Even) = 1/2$. Let $n$ be the number of trials.
Step 2: $^nC_2 (1/2)^n = ^nC_3 (1/2)^n \Rightarrow ^nC_2 = ^nC_3 \Rightarrow n = 2+3=5$.
Step 3: Probability of odd number of successes in $n$ trials with $p=1/2$: $P = \frac{1}{2^n} \left[ ^nC_1 + ^nC_3 + ^nC_5 + ....... \right]$.
Step 4: Since the sum of odd binomial coefficients is $2^{n-1}$, $P = \frac{2^{n-1}}{2^n} = \frac{1}{2}$.