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The Chess Problem. This year’s Belmont chess champion is to be selected by the following
procedure. Bo and Ci, the leading challengers, first play a two-game match. If one of them wins
both games, he gets to play a two-game second round with Al, the current champion. Al retains
his championship unless a second round is required and the challenger beats Al in both games.
If Al wins the initial game of the second round, no more games are played.
Furthermore, we know the following:
• The probability that Bo will beat Ci in any particular game is 0.6.
• The probability that Al will beat Bo in any particular game is 0.5.
• The probability that Al will beat Ci in any particular game is 0.7.
Assume no tie games are possible and all games are independent.
(a) Determine the apriori probabilities that
i. the second round will be required.
ii. Bo will win the first round.
iii. Al will retain his championship this year.

1) Probability the second round happens enter image description here

2) Probability Bo wins enter image description here

3) Al retains championship enter image description here

However I decided to solve the 3rd differently... by considering that Al could only loose her championship if second round happens and then multiplying its probability with probabilities of Bo or Ci winning. However I reach at wrong answer (and this also doesn't make sense algebraically) But I still can't intuitively understand why this is not the case. enter image description here

3 Answers 3

1

You problem in the last formulation there is that you are effectively allowing the probability that the person who loses the first round, beats Al in the second round. The chance of there being a second round includes two victories by either player

You first, correct, diagram for question 3 would be modified to have an addition node after the first two games, but obviously that doesn't happen.

1

Your last diagram neglects the fact that Bo and Ci have different chances of reaching the second round. You have to weight the upper and the lower branch of the diagram according to these chances as it is done in the first diagram for the third question.

0

It seems to me that given that a second round happens, the probability that Bo wins is not 50%. Bo might not be in the second round.