Question

Find the probability of each event. A gambler places a bet on a horse race. To win, he must pick the top three finishers in any order. Eight horses of equal ability are entered in the race. Assuming the horses finish in a random order, what is the probability that the gambler will win his bet?

Answer 1

The probability that the gambler will win his bet is 1 or 100%.

Step-by-step explanation:

To calculate the probability of the gambler winning his bet, we need to determine the total number of possible outcomes and the number of favorable outcomes. In this case, the total number of possible outcomes is the number of ways the top three finishers can be arranged out of the eight horses. This can be calculated using the formula for combinations:

C(n, r) = n! / (r!(n-r)!)

For this problem, n = 8 (number of horses) and r = 3 (number of finishers). Plugging these values into the formula, we get:

C(8, 3) = 8! / (3!(8-3)!) = 56

So there are 56 possible outcomes.

Now, let's determine the number of favorable outcomes. Since the order of the finishers doesn't matter for the gambler's bet, we need to calculate the number of ways to choose three horses out of the eight without considering their order. This can be calculated using the formula for combinations:

C(n, r) = n! / (r!(n-r)!)

For this problem, n = 8 (number of horses) and r = 3 (number of finishers). Plugging these values into the formula, we get:

C(8, 3) = 8! / (3!(8-3)!) = 56

So there are 56 favorable outcomes.

Finally, we can calculate the probability of the gambler winning his bet by dividing the number of favorable outcomes by the total number of outcomes:

Probability = favorable outcomes / total outcomes = 56 / 56 = 1

Therefore, the probability that the gambler will win his bet is 1 or 100%.