SOLUTION: Alice and Bob play three chess games. Alice is 10 times more likely to win a game than Bob. Find the probability that
Alice will win all three games.
AND
Bob will win all
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Alice will win all three games.
AND
Bob will win all
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Question 1175734: Alice and Bob play three chess games. Alice is 10 times more likely to win a game than Bob. Find the probability that
Alice will win all three games.
AND
Bob will win all three games. Found 2 solutions by ikleyn, greenestamps:Answer by ikleyn(52787) (Show Source):
The translated, the condition means that
p(A wins a game) = ; p(B wins a game) = .
THEREFORE,
(a) p(Alice wins all 3 games of 3) = = 0.7513 = 75.13%.
(b) p(Bob wins all 3 games of 3) = = 0.000751 = 0.075%.
The language used in the statement of the problem is poor, leading to different interpretations and different answers.
Tutor @ikleyn interpreted "Alice is 10 times more likely to win a game than Bob" to mean that if Bob's probability is x then Alice's probability is 10 times x, or 10x. That makes their probabilities of winning 1/11 and 10/11.
That is a possible interpretation of the given information.
However, the grammatically correct interpretation of the phrase is that if Bob's probability is x then Alice's probability is x PLUS 10 (more) times x, or x+10x = 11x. That makes their probabilities of winning 1/12 and 11/12.
So my answers to the questions would be
P(Alice wins all 3 games) = (11/12)^3
P(Bob wins all 3 games) = (1/12)^3
Unfortunately, in everyday usage "4 times as much as" and "4 times more than" are carelessly used to mean the same thing. And to further the misfortune, the "4 times more than" phrase is used far more often than the "4 times as much as" phrase; so most of the time the information that is supposed to be getting conveyed is false.
It is very likely that the author of the problem intended the answers to be given as shown by the other tutor. However, if the statement of the problem is interpreted grammatically correctly, the correct answers are those that I show.
