Question

Racehorse. A man buys a racehorse for \(\$ 20,000\) and enters it in two races. He plans to sell the horse afterward, hoping to make a profit. If the horse wins both races, its value will jump to \(\$ 100,000\). If it wins one of the races, it will be worth \(\$ 50,000\). If it loses both races, it will be worth only \(\$ 10,000\). The man believes there's a \(20 \%\) chance that the horse will win the first race and a \(30 \%\) chance it will win the second one. Assuming that the two races are independent events, find the man's expected profit.

Short answer

The expected profit is $10,600.

Step-by-step solution

Determine the Probability of Winning Both Races

The probability of winning both the first and second race is given by multiplying the probabilities of winning each race. Since both races are independent, this probability is:\[P(\text{Win both}) = P(\text{Win first}) \times P(\text{Win second}) = 0.20 \times 0.30 = 0.06\]So, there is a 6% chance the horse will win both races.

Determine the Probability of Winning Only One Race

There are two possible scenarios for winning only one race: winning the first race and losing the second, or losing the first race and winning the second. The probabilities for these scenarios are:\[P(\text{Win first, lose second}) = 0.20 \times (1 - 0.30) = 0.20 \times 0.70 = 0.14\]\[P(\text{Lose first, win second}) = (1 - 0.20) \times 0.30 = 0.80 \times 0.30 = 0.24\]Adding these probabilities gives the chance of winning one race:\[P(\text{Win one race}) = 0.14 + 0.24 = 0.38\]

Determine the Probability of Losing Both Races

The probability of losing both races is found by multiplying the chances of losing each race:\[P(\text{Lose both}) = (1 - 0.20) \times (1 - 0.30) = 0.80 \times 0.70 = 0.56\]This means there is a 56% chance the horse will lose both races.

Calculate Expected Value of Final Horse Value

Using the probabilities calculated in previous steps, multiply each outcome by its probability and sum them to find the expected value.\[E(\text{Final value}) = P(\text{Win both}) \times 100,000 + P(\text{Win one}) \times 50,000 + P(\text{Lose both}) \times 10,000\]\[E(\text{Final value}) = 0.06 \times 100,000 + 0.38 \times 50,000 + 0.56 \times 10,000\]\[E(\text{Final value}) = 6,000 + 19,000 + 5,600 = 30,600\]

Calculate Expected Profit

Subtract the purchase price of the horse from the expected value to find the expected profit.\[E(\text{Profit}) = E(\text{Final value}) - \text{Purchase price} = 30,600 - 20,000 = 10,600\]The expected profit from buying, racing, and selling the horse is $10,600.

Key concepts

Expected Value

Expected value is a fundamental concept in probability theory that helps us understand what to anticipate over the long term when dealing with random variables. In the context of the racehorse example, the expected value provides an average outcome of the horse's final value after racing. It does this by weighing each possible outcome by its probability.
To compute the expected value for the final value of the horse:

• Calculate the probability for each event: winning both races, winning just one race, or losing both.
• Multiply these probabilities by their respective outcomes (values).
• Add the results to find the overall expected value.

This approach gives you a sense of what the horse is "worth" on average after racing, which in this exercise amounted to $30,600. Understanding expected value is key to making informed decisions in uncertain scenarios.

Independent Events

In probability, events are considered independent if the outcome of one does not affect the outcome of another. This idea simplifies probability calculations because you can multiply the probabilities of independent events to find the overall probability.
In the racehorse example, the races are deemed independent events. This means:

• The chance of the horse winning the second race is not influenced by its result in the first race, and vice versa.
• This independence allows for straightforward calculations using basic probability rules.

The calculation of the horse's probability of winning both races was a direct product of the independent probabilities: 20% for the first race and 30% for the second, resulting in a 6% chance of winning both. Understanding independence is crucial for calculating the joint probabilities of multiple events.

Probability Calculations

Probability calculations involve determining the likelihood that various outcomes will occur. These calculations are foundational in assessing risks and expected outcomes in uncertain situations, like the horse racing scenario.
Here's how you perform these calculations:

• Use multiplication to find the probability of two independent events happening together. For example, winning both races.
• Use addition when considering the probability of mutually exclusive events—different event outcomes that can't happen at the same time, like winning only one race.

For the exercise, the probability of winning one race was found by considering two scenarios: either the horse wins the first race and loses the second, or it loses the first but wins the second. By calculating individually and summing, the result was a 38% chance of winning one race. Properly calculating probabilities is essential for predicting likely outcomes and making strategic decisions under uncertainty.