JEE Main · 2021 · 24 Feb 2021 · Shift 1 · Probability

With a fair die the chances of an odd face and of an even face are each $\frac{1}{2}$, so for $n$ throws the probabilities of two odd throws and of three even throws are $\left(\genfrac{}{}{0ex}{}{n}{2}\right){\left(\frac{1}{2}\right)}^n$ and $\left(\genfrac{}{}{0ex}{}{n}{3}\right){\left(\frac{1}{2}\right)}^n$. Equating them forces $\left(\genfrac{}{}{0ex}{}{n}{2}\right)=\left(\genfrac{}{}{0ex}{}{n}{3}\right)$, which holds when $2+3=n$, that is $n=5$. For a binomial experiment with success probability $\frac{1}{2}$ the chance of an odd number of successes equals that of an even number, so the probability of obtaining an odd face an odd number of times is $\frac{\left(\genfrac{}{}{0ex}{}{5}{1}\right)+\left(\genfrac{}{}{0ex}{}{5}{3}\right)+\left(\genfrac{}{}{0ex}{}{5}{5}\right)}{2^5}=\frac{5+10+1}{32}=\frac{16}{32}=\frac{1}{2}$.