Let's describe three different sample spaces based on the outcomes of the horse race according to the given scenarios:

a. You bet on a single horse and care whether you win or lose.

In this case, the sample space consists of just two possible outcomes: whether your horse wins or loses.

• Sample Space: $S = \{ \text{Win}, \text{Lose} \}$

b. You care which of the five horses wins.

Here, the sample space consists of all possible outcomes where only one horse wins the race. Since there are 5 horses, there are 5 possible outcomes, one for each horse.

• Sample Space: $S = \{ H_1, H_2, H_3, H_4, H_5 \}$ where $H_1, H_2, H_3, H_4, H_5$ represent the five horses, and the outcome is whichever horse wins.

c. You care about which horses come in first, second, and third.

In this case, the sample space is more complex because the order of the top three horses matters. This is a permutation problem where you are selecting 3 horses out of 5, with the order being important.

• Sample Space: The total number of outcomes (permutations) is: $5P3 = \frac{5!}{(5-3)!} = 5 \times 4 \times 3 = 60$ Therefore, the sample space consists of 60 different ordered triplets representing the horses that finish in 1st, 2nd, and 3rd place. Each outcome can be written as $(H_i, H_j, H_k)$ where $H_i$, $H_j$, and $H_k$ are distinct horses.

• Sample Space Example: $S = \{ (H_1, H_2, H_3), (H_1, H_2, H_4), \dots, (H_5, H_4, H_3) \}$

These sample spaces represent different levels of detail depending on what aspect of the race you are interested in.

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Would you like more details on any of these sample spaces or have any specific questions?

Here are 5 related questions for deeper exploration:

1. What is the probability of winning if you randomly bet on one horse?
2. How would the sample space change if the race had 6 horses instead of 5?
3. Can you explain how to calculate probabilities based on the sample space in part c?
4. How many possible outcomes are there if you care about the horses that place 1st and 2nd?
5. What is the difference between combinations and permutations in this context?

Tip: Always remember that order matters in permutations but not in combinations.