JEE Main 2026 (Online) 2nd April Evening ShiftQuestion 1
A man throws a fair coin repeatedly. He gets $10$ points for each head he throws and $5$ points for each tail he throws. If the probability that he gets exactly $30$ points is $\frac{m}{n}$, $\gcd(m,n)=1$, then $m+n$ is equal to :
Probability
Solution
Let $f(s)$ = P(total hits exactly $s$). $f(s)=\frac{f(s-10)+f(s-5)}{2}$, $f(0)=1$. Equivalently by cases: 3H: $\frac18$; 2H2T: $\binom42/2^4=\frac{24}{64}$; 1H4T: $5/32=\frac{10}{64}$; 6T: $\frac1{64}$. Sum $=\frac{8+24+10+1}{64}=\frac{43}{64}$, so $m+n=107$. Option (C).