SOLUTION: The probability that an athlete will not win any of the three races is 1/4. If the athlete runs in all the races, what is the probability that the athlete will win ( i) only the

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Question 1195406: The probability that an athlete will not win any of the three races is 1/4. If the athlete runs in all the races, what is the probability that the athlete will win ( i) only the second race (ii) all the three races (iii) only two of the races
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52784)   (Show Source): You can put this solution on YOUR website!
.
The probability that an athlete will not win any of the three races is 1/4.
If the athlete runs in all the races, what is the probability
that the athlete will win
(a) only the second race
(b) all the three races
(c) only two of the races
~~~~~~~~~~~~~~~~~~~~~


            From the context, the probability for the athlete to win any of the three
            races is 3/4, as a complement to the given probability not to win of 1/4.

            Now it is easy to answer all questions, one after the other. 


(a)  P = P(lose,win,lose) =  = .    ANSWER


(b)  P = P(win,win,win) =  = .             ANSWER


(c)  P(win only 2 of 3 races) = use the formula of binomial distribution 
                                for 3 trials, 2 success and individual probability of success p = 3/4

     = P(n=3, k=2, p=3/4) =  =  =  = .      ANSWER


Incidentally,  (b)  and  (c)  have equal probabilities.

Solved.



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


"The probability that an athlete will not win any of the three races is 1/4."

In the solution from tutor @ikleyn, she says that "from the context..." the meaning of the given information is that the probability of his not winning in EACH of the three races is 1/4.

But that is not what the given information says.

It might be what the given information was SUPPOSED TO say; and indeed the questions that are asked can only be answered if that is what it was supposed to say.

But the given information only tells us that the probability is 1/4 that the number of races he wins is 0; and from that given information we can only conclude that there is a 3/4 probability that he will win at least one of the three races.

And the questions cannot be answered with only that information.

My conclusion:

The probability is about 99% that the problem was stated incorrectly.
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