Probability Race Horse

Question

A horse race has 10 entries and one person owns 4 of those horses. Assuming that there are no ties, what is the probability that those four horses finish first, second, third, and fourth?

Answer 1

Another way to get the answer is to find out how many groups of 4 horses can one pick from 10 horses. Then knowing that only one group contains all 4 or our owners horses, we can come up with the probability.

To find the number of groups we can make of 4 horses out of 10 horses, we use the formula for combinations

10! Ten Factorial

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4! x 6! Four Factorial times Six Factorial 10 x 9 x 8 x 7 / 4 x 3 x 2 x 1 = 210

When we calculate the answer, we find there are 210 ways to combine groups of 4 horses out of 10.

Since there is only 1 group that has all four of the owner's horses, the probability is 1/210

Answer 2

Assuming random finishes, the probability that one of the owner's horses wins first place is 4/10 = 2/5.

Once that happens, the probability that one of the owner's horses wins second place is 3/9 = 1/3. Once that finishes, the probability that one of the owner's horses will win third place is 2/8 = 1/4. And then lastly, the probability that the last of the owner's horses will win 4th place is 1/7.

So the probability that all 4 events occur is 2/5 • 1/3 • 1/4 • 1/7 = 1/210