Question 495284: I have 8 runners in a race...what is the probability that i will pick 1, 2 and 3rd place
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the probability you will pick the winner is 1/8
if you are so lucky to pick the winner, the probability you will pick second place runner is 1/7.
if you are so lucky to pick second place, the probability you will pick third place is 1/6.
the probability you will pick all 3 is 1/8 * 1/7 * 1/6 = 1/336.
you might be able to see this easier if we had only 3 runners and we had to pick the first and second place finish.
assume the runners are a, b, and c
the probability you will pick the winner is 1 out of 3.
the possible finishes are:
abc
acb
bac
bca
cab
cba
you can see that:
a finishes first in 2 out of the 6 possible permutations.
b finishes first in 2 out of the 6 possible permutations.
c finishes first in 2 out of the 6 possible permutations.
the probability of any one of them finishing first is 2 out of 6 which is equal to 1/3.
now, suppose you picked a, and a was in first place.
that leave b and c to finish second or third.
the possible permutations are now:
bc
cb
that means that:
b finishes second in 1 out of the 3 possible permutations.
c finishes second in 1 out of the 3 possible permutations.
the probability of any one of them finishing second is 1 out of 2 which is equal to 1/2.
the probability of both occurrences happening is 1/3 * 1/2 = 1/6.
look at all 6 possible permutations again.
abc
acb
bac
bca
cab
cba
you pick a to be the winner and b to be second.
a is the winner and b is second in only 1 out of the 6 possible permutations.
that's your answer.
the probability is 1 out of 6.
in your problem, the number of runners is 8 and you are trying to pick the first 3 winners.
p(first) is 1/8
p(second) is 1/7
p(third) is 1/6
p(all three) is 1/8 * 1/7 * 1/6 = 1/336
same thing only more runners to choose from so it's a lot harder to show you how it works.
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