If the odds in favor of Fast Leg winning a horse race are 9 to 7 and the first prize is \( \$ 112,000 \), what is the expected value of Fast Leg's winnings? The expected value is \( \$ \square \).

To calculate the expected value of Fast Leg's winnings, we need to consider both the likelihood of winning and losing. The odds of 9 to 7 mean that for every 16 races (9 wins + 7 losses), Fast Leg is expected to win 9 times. Therefore, the probability of winning is \( \frac{9}{16} \) and the probability of losing is \( \frac{7}{16} \). If Fast Leg wins, the winnings are \( \$ 112,000 \). If Fast Leg loses, the winnings are \( \$ 0 \). The expected value (EV) can be calculated as follows: \[ EV = \left( \text{Probability of Winning} \times \text{Winnings} \right) + \left( \text{Probability of Losing} \times \text{Losses} \right) \] \[ EV = \left( \frac{9}{16} \times 112,000 \right) + \left( \frac{7}{16} \times 0 \right) = \left( \frac{9 \times 112,000}{16} \right) \] \[ EV = \frac{1,008,000}{16} = 63,000 \] Thus, the expected value of Fast Leg's winnings is \( \$ 63,000 \).