To solve this problem, we need to calculate the probability of winning the bet, which requires randomly selecting 3 horses that finish in the first, second, and third places in any order.

Step 1: Total Number of Outcomes

The total number of ways to select 3 horses out of 17 is represented by a combination since the order doesn't matter when selecting the horses. The formula for a combination is:

$C(n, r) = \frac{n!}{r!(n - r)!}$

Where:

• $n = 17$ (the total number of horses),
• $r = 3$ (the number of horses to be selected).

So, the total number of ways to select 3 horses out of 17 is:

$C(17, 3) = \frac{17!}{3!(17 - 3)!} = \frac{17 \times 16 \times 15}{3 \times 2 \times 1} = 680$

Step 2: Number of Favorable Outcomes

To win the bet, you need to correctly select the 3 horses that finish in the first, second, and third places. Since the order of finishing matters, we are essentially selecting an ordered combination (or permutation).

The number of favorable outcomes (i.e., the number of ways the 3 horses can finish in the first, second, and third positions) is simply the number of ways to arrange the 3 horses in these positions. This is a permutation of 3 horses:

$P(3) = 3! = 3 \times 2 \times 1 = 6$

Step 3: Calculating the Probability

The probability of winning is the ratio of favorable outcomes to total outcomes:

$P(\text{win}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{6}{680} \approx 0.00882$

Final Answer:

The probability of winning the bet is approximately 0.00882 (rounded to five decimal places).

Let me know if you have any further questions or need clarification!

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Related Questions:

1. What is the probability of winning if you select 2 horses instead of 3?
2. How does the probability change if there are more horses in the race?
3. What is the formula for calculating permutations and how does it differ from combinations?
4. How can we use the combination formula for problems involving selection without regard to order?
5. If there are 20 horses in a race, how many combinations can you make by choosing 3 horses?

Tip: For probability problems involving selections, always identify whether the order matters. If the order does matter, use permutations; if not, use combinations!