13)   The odds against a particular horse winning the Kentucky Derby is 7:4.  What is the probability the horse wins?

To determine the probability that the horse wins, we need to understand the relationship between odds and probability.

1. The odds against the horse winning are given as 7:4. This means that for every 7 times the horse does not win, it wins 4 times.

2. To convert these odds into a probability, we need to find the total number of possible outcomes. The total number of outcomes is the sum of the two parts of the odds: 7 (not winning) + 4 (winning) = 11.

3. The probability of the horse winning is the number of winning outcomes divided by the total number of outcomes. Therefore, the probability is 4 (winning outcomes) divided by 11 (total outcomes).

4. So, the probability that the horse wins is: $\text{Probability} = \frac{4}{11}$

Thus, the probability that the horse wins the Kentucky Derby is $\frac{4}{11}$.