JEE Main · 2026 · 23 Jan 2026 · Shift 2 · Probability

Condition on the ball moved from $B$ into $A$. A white transfer has probability $\frac{6}{10}=\frac{3}{5}$, after which $A$ holds $10$ white and $8$ black, giving a white draw with probability $\frac{10}{18}=\frac{5}{9}$. A black transfer has probability $\frac{4}{10}=\frac{2}{5}$, after which $A$ holds $9$ white and $9$ black, giving a white draw with probability $\frac{1}{2}$. The total probability is $\frac{3}{5}\cdot \frac{5}{9}+\frac{2}{5}\cdot \frac{1}{2}=\frac{1}{3}+\frac{1}{5}=\frac{8}{15}$. With $\gcd (8,15)=1$, $p+q=8+15=23$.