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Chapter 16: Problem 8

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

Expert verified
The expected profit is $10,600.

Step by step solution

01

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.
02

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:\[P(\text{Win one race}) = 0.14 + 0.24 = 0.38\]
03

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.
04

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.\[E(\text{Final value}) = P(\text{Win both}) \times 100,000 + P(\text{Win one}) \times 50,000 + P(\text{Lose both}) \times 10,000\]\[E(\text{Final value}) = 0.06 \times 100,000 + 0.38 \times 50,000 + 0.56 \times 10,000\]\[E(\text{Final value}) = 6,000 + 19,000 + 5,600 = 30,600\]
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攄ifferent 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.

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Most popular questions from this chapter

Cereal. The amount of cereal that can be poured into a small bowl varies with a mean of \(1.5\) ounces and a standard deviation of \(0.3\) ounces. A large bowl holds a mean of \(2.5\) ounces with a standard deviation of \(0.4\) ounces. You open a new box of cereal and pour one large and one small bowl. a) How much more cereal do you expect to be in the large bowl? b) What's the standard deviation of this difference? c) If the difference follows a Normal model, what's the probability the small bowl contains more cereal than the large one? d) What are the mean and standard deviation of the total amount of cereal in the two bowls? e) If the total follows a Normal model, what's the probability you poured out more than \(4.5\) ounces of cereal in the two bowls together? f) The amount of cereal the manufacturer puts in the boxes is a random variable with a mean of \(16.3\) ounces and a standard deviation of \(0.2\) ounces. Find the expected amount of cereal left in the box and the standard deviation.

Insurance. An insurance policy costs \(\$ 100\) and will pay policyholders \(\$ 10,000\) if they suffer a major injury (resulting in hospitalization) or \(\$ 3000\) if they suffer a minor injury (resulting in lost time from work). The company estimates that each year 1 in every 2000 policyholders may have a major injury, and 1 in 500 a minor injury only. a) Create a probability model for the profit on a policy. b) What's the company's expected profit on this policy? c) What's the standard deviation?

Fire! An insurance company estimates that it should make an annual profit of \(\$ 150\) on each homeowner's policy written, with a standard deviation of \(\$ 6000\). a) Why is the standard deviation so large? b) If it writes only two of these policies, what are the mean and standard deviation of the annual profit? c) If it writes 10,000 of these policies, what are the mean and standard deviation of the annual profit? d) Is the company likely to be profitable? Explain. e) What assumptions underlie your analysis? Can you think of circumstances under which those assumptions might be violated? Explain.

Kittens. In a litter of seven kittens, three are female. You pick two kittens at random. a) Create a probability model for the number of male kittens you get. b) What's the expected number of males? c) What's the standard deviation?

Casino. A casino knows that people play the slot machines in hopes of hitting the jackpot but that most of them lose their dollar. Suppose a certain machine pays out an average of \(\$ 0.92\), with a standard deviation of \(\$ 120 .\) a) Why is the standard deviation so large? b) If you play 5 times, what are the mean and standard deviation of the casino's profit? c) If gamblers play this machine 1000 times in a day, what are the mean and standard deviation of the casino's profit? d) Is the casino likely to be profitable? Explain.
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