0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Please explain this: If someone wins the match, that is scores 3 points then the one scores 3 points independently of the length of the match. $\endgroup$ – zoli Mar 11 '18 at 15:00
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.