There are 3(A,B,C) participants participating in the competition. The first question is asked to A, next one to B and the next to C and this repeats. There are 3 ways in which an answer is scored, wrong guess: 1 point, partial answer: 2 points and correct answer: 3 points. The person who scores 4 points or above first wins the competition. The probability of scoring 1, 2 or 3 points for a question is same and also the same for every participant. Find the probability that A wins the competition if a question with a wrong guess or partial answer doesn't pass on to the others.
Any given participant, if allowed to continue, would achieve four or more points by taking at least 2 turns and at most 4. A's advantage is that he goes first and those who haven't achieved 4 when one person gets 4 don't get "last licks"--there are no ties.
Going back to the individual participant achieving the goal in 2 to 4 rounds:
It can be achieved in 2 rounds by getting:
1 and 3 prob 1/9
2 and either 2 or 3 prob 2/9
3 and anything prob 1/3
At the other end of the luck (or skill) scale is taking 4 turns to get to the goal. It requires getting 1 point in each of the first three rounds and anything in the fourth.
This has probability 1/27.
That leaves the probability of achieving the goal on the third round as 1 - 2/3 - 1/27 = 8/27.
Back to having three participants:
How can A win?
He can achieve the goal on round 2. This doesn't give any of the others a chance, so it has probability 2/3, the same as if he were playing alone.
He can achieve the goal in round 3 after neither of his opponents has succeeded in round 2. The probability that both opponents will have failed in round 2 is (1/3)^2 = 1/9. A's probability of 8/27 of hitting his goal on round 3 already includes the consideration of his having failed by round 2, so it's the only other number we need. The probability, in the overall scheme, that A will win against his competition at round 3 is (1/9)*(8/27) = 8/243.
The remaining possibility for A's win is that each of the three competitors takes the full four rounds to reach the goal. That has probability (1/27)^3 = 1/19683.
That makes A's overall probability of winning the competition 2/3 + 8/243 + 1/19683 = 13771/19683, a number that has a very long period in decimal representation:
where the ellipsis indicates repetition starting again with the 699639 of the beginning.
Here are the results of 3 million trials (one million at a time, three times):
A wins B wins C wins
700155 225907 73938
699798 225898 74304
699766 225901 74333
This agrees with the calculated probability for A, and also shows that B has about a 22.6% probability of winning and C has about a 7.4% probability.
FOR trial = 1 TO 1000000
FOR i = 1 TO 3
pt = INT(3 * RND(1) + 1)
pts(i) = pts(i) + pt
IF pts(i) >= 4 THEN winCt(i) = winCt(i) + 1: EXIT DO
FOR i = 1 TO 3
Posted by Charlie
on 2009-01-25 19:47:58