Let's solve the problem step-by-step.

Step 1: Define the possible outcomes

We have three possible outcomes for the horse's performance in the two races:

1. The horse wins both races: The horse's value increases to $80,000.
2. The horse wins exactly one race: The horse's value increases to $50,000.
3. The horse loses both races: The horse's value decreases to $15,000.

Step 2: Determine the probabilities of each outcome

We are given the following probabilities for the races:

• The probability that the horse wins the first race is 25% (or 0.25).
• The probability that the horse wins the second race is 40% (or 0.40).
• The two events are independent, so we can use the multiplication rule for independent events to calculate the probabilities of each outcome:

Outcome 1: Wins both races

• Probability = $0.25 \times 0.40 = 0.10$

Outcome 2: Wins exactly one race

This occurs in two sub-cases:

• Wins the first race and loses the second: $0.25 \times (1 - 0.40) = 0.25 \times 0.60 = 0.15$
• Loses the first race and wins the second: $(1 - 0.25) \times 0.40 = 0.75 \times 0.40 = 0.30$

The total probability of winning exactly one race is:

• $0.15 + 0.30 = 0.45$

Outcome 3: Loses both races

• Probability = $(1 - 0.25) \times (1 - 0.40) = 0.75 \times 0.60 = 0.45$

Step 3: Determine the corresponding profits

We can now calculate the profit for each outcome:

• Wins both races: The horse is sold for $80,000. The profit is $80,000 - 20,000 = 60,000$.
• Wins exactly one race: The horse is sold for $50,000. The profit is $50,000 - 20,000 = 30,000$.
• Loses both races: The horse is sold for $15,000. The profit is $15,000 - 20,000 = -5,000$.

Step 4: Calculate the expected profit

The expected profit is the sum of the products of each possible profit and its corresponding probability:

$\text{Expected Profit} = (60,000 \times 0.10) + (30,000 \times 0.45) + (-5,000 \times 0.45)$

$\text{Expected Profit} = 6,000 + 13,500 - 2,250 = 17,250$

Final Answer:

The expected profit is $17,250.

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Would you like more details or have any questions about the solution? Here are some related questions for further exploration:

1. How would the expected profit change if the probability of winning the first race were increased to 50%?
2. What would happen to the expected profit if the value of the horse after losing both races increased to $25,000?
3. If the cost of the horse was $25,000 instead of $20,000, how would this affect the expected profit?
4. How can we calculate the expected value of an event with three possible outcomes?
5. What is the difference between independent and dependent events in probability theory?

Tip: Remember that when calculating the expected value, it's important to account for all possible outcomes and their probabilities, and multiply each profit by its associated probability.