Let $X$ denote the number of points scored by a chess player in a match against an equally strong opponent, where draws are discarded so that the odds of winning or losing a game are fifty fifty. The match ends if one of the players reaches 3 points, and $B_i$ is the event that $i$ games have been played. Then find the conditional expectation $E(X|B_i)$.
Let a player win $p (0<p<3)$ points on winning a game and $0$ on losing one. If a player, with greater points than his opponent, has won $j$ games $(0 \leq j \leq i)$ then we can calculate $P(\mathbb I_{B_i}X)$=$i \choose j$$( \frac{1}{2^i})$ and therefore, $E(\mathbb I_{B_i}X)=\frac{(2^i-1)ip}{2^i}$. We note that $jp < 3.$
But, I am having trouble calculating $P(B_i).$ Kindly provide your valuable input.
Hint: For $j=1,2,3$ the number of points scored by the player after $j$ games is $C_j$, where $C_j \sim Bin(j, 1/2)$. Then $E(X|B_j)=E(C_j)$.
The case $B_5$ means that one player scored 2 and the other scored 3. $B_4$ is similar, it might help to write out the possible combinations and compute it that way.