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Chapter 7: Problem 249

The State of New Hampshire conducts an annual lottery to raise funds for the school districts in the state. Assume a million tickets are sold. One ticket is the winning ticket and the winner receives \(\$ 10,000\). If each ticket costs \(\$ .25\), find the expected value of a randomly purchased ticket and the revenue that the lottery generates for the school districts in the state.

Short Answer

Expert verified
The expected value of a randomly purchased lottery ticket is \(\$0.10\) and the total revenue generated by the lottery for the school districts in the state is \(\$240,000\).

Step by step solution

01

Define probabilities and values

In this lottery, there are two possible outcomes: winning the \(\$10,000\) prize or not winning (losing). Since there are a million tickets and only one winning ticket, the probability of winning is \(\frac{1}{1,000,000}\) and the probability of losing is \(\frac{999,999}{1,000,000}\). The value corresponding to each outcome is as follows: - Winning: \(\$10,000\) - Losing: \(\$0\) (since there is no smaller prize for losing)
02

Calculate the expected value of a ticket

To compute the expected value of a single ticket, we need to multiply the value of each outcome by its probability and then add them together. Therefore, the expected value E(x) can be calculated as follows: E(x) = (Winning value 脳 Probability of winning) + (Losing value 脳 Probability of losing) E(x) = (\(\$10,000\) 脳 \(\frac{1}{1,000,000}\)) + (\(\$0\) 脳 \(\frac{999,999}{1,000,000}\)) E(x) = \(\$10\) 脳 \(\frac{1}{100}\) E(x) = \(\$0.10\) So the expected value of each ticket is \(\$0.10\).
03

Calculate the total revenue from the lottery

To find the total revenue generated by the lottery, we'll subtract the winning amount paid out to the winner from the total amount collected by selling the tickets. First, let's find the amount collected from selling all the tickets: Amount collected = Number of tickets 脳 Price per ticket Amount collected = \(1,000,000\) 脳 \(\$0.25\) Amount collected = \(\$250,000\) Now let's subtract the winning amount: Revenue = Amount collected - Winning amount Revenue = \(\$250,000\) - \(\$10,000\) Revenue = \(\$240,000\) Thus, the total revenue generated by the lottery for the school districts in the state is \(\$240,000\). In conclusion, the expected value of a randomly purchased lottery ticket is \(\$0.10\), and the revenue generated by the lottery for the school districts in the state is \(\$240,000\).

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Key Concepts

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

Understanding Probability
Probability is a fundamental concept in mathematics, expressing the likelihood that a particular event will occur. It is often represented as a number between 0 and 1, where 0 means the event is impossible and 1 means it is certain. In our lottery example, we have two primary outcomes: winning the prize or not winning. The probability of winning is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. Here, there is one winning ticket in a million, so the probability of winning is
  • Probability of winning: \( \frac{1}{1,000,000} \)
  • Probability of losing: \( \frac{999,999}{1,000,000} \)
These probability values help us to understand the chances of different outcomes, which is essential for calculating expected value.
Calculating Lottery Revenue
When calculating lottery revenue, we consider the total amount collected from ticket sales and the subtraction of any prizes paid out. The total revenue from the lottery is the amount left after paying the prize money. In our example:
  • Total amount collected from ticket sales: \( 1,000,000 \times \\(0.25 = \\)250,000 \)
  • Prize money paid: \\(10,000
Thus, the revenue generated by the lottery for school funding is the total sales minus the prize money, which is \( \\)250,000 - \\(10,000 = \\)240,000 \). This remaining sum is what the schools can use for funding improvement projects. By understanding how revenue is calculated, students can see the importance of responsible lottery structuring.
Impact of Lottery Revenue on School Funding
Lottery revenue can have a significant impact on school funding, providing schools with financial resources that might otherwise be unavailable. States often use lotteries as a means to support educational programs, infrastructure improvements, and new projects without raising taxes. In our example, the revenue earned from the lottery is \( \$240,000 \), a substantial amount that can be directed toward schools in need. This funding can be used for
  • Updating classroom technology
  • Hiring additional staff
  • Improving facilities
By investing in education, the lottery supports better learning environments and opportunities for students. This demonstrates how a well-orchestrated lottery can benefit the entire community.
Outcome Probabilities and Expected Value
Outcome probabilities directly influence an important statistical concept known as expected value. Expected value is crucial in decision-making, as it represents the average outcome one can expect from a particular scenario over a long time. To find the expected value of a lottery ticket, you multiply each outcome's value by its probability and then sum them. Here's how it's done:
  • Expected value, \( E(x) = (\\(10,000 \times \frac{1}{1,000,000}) + (\\)0 \times \frac{999,999}{1,000,000}) \)
  • This simplifies to \( E(x) = \\(0.10 \)
This means that, on average, a lottery ticket is worth just \\)0.10, despite costing \$0.25. Understanding expected value helps individuals evaluate whether the risk of purchasing a ticket is worth the potential reward, balancing the probability of winning against the game's cost.

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Most popular questions from this chapter

Suppose the earnings of a laborer, denoted by \(\mathrm{X}\), are given by the following probability function. \begin{tabular}{|c|c|c|c|c|} \hline \(\mathrm{X}\) & 0 & 8 & 12 & 16 \\ \hline \(\operatorname{Pr}=(\mathrm{X}=\mathrm{x})\) & \(0.3\) & \(0.2\) & \(0.3\) & \(0.2\) \\ \hline \end{tabular} Find the laborer's expected earnings.

The lifetime \(\mathrm{x}\) (in hours) of electronic tubes mass-produced by a standard process is a random variable with a probability distribution having the density function \(\alpha^{2} \mathrm{xe}^{-\alpha \mathrm{x}}\) for \(\mathrm{x} \geq 0\) Prove that the expected lifetime of the tubes is \(2 / \alpha\). Also, show that the probability that a random selected tube will have a lifetime \(\geq \mathrm{m}\) is \((1+\alpha \mathrm{m}) \mathrm{e}^{-\alpha \mathrm{m}}\). A research engineer suggests certain modifications in the production process which would alter the lifetime distribution by increasing the expected lifetime to \(2 / \beta(\beta<\alpha) .\) But because of the cost of introducing the change the manufacturer does not consider it worth while unless the modified process also ensures that the probability of the lifetime of a randomly selected tube being \(\geq \mathrm{m}\) is increased by a fraction \(\lambda(>0)\). Prove that the manufacturer's condition for introducing the new process is satisfied if $$ \beta<\alpha-(1 / \mathrm{m}) \log _{\mathrm{e}}(1+\lambda) $$

Show that the sum of the expected value of two discrete random variables with joint density \(\mathrm{f}(\mathrm{x}, \mathrm{y})\) is equal to the expected value of the sum of these two random variables. That is \(\mathrm{E}(\mathrm{X}+\mathrm{Y})=\mathrm{E}(\mathrm{X})+\mathrm{E}(\mathrm{Y})\)

Compute the conditional distribution of \(\mathrm{Y}\) given \(\mathrm{X}\) if \(\mathrm{X}\) and \(\mathrm{Y}\) are jointly distributed with density \(f(x, y)=x+y\) $$ 0<\mathrm{x}<1 $$ $$ 0

A new insecticide is advertised as being \(90 \%\) effective in killing ants with 1 application. If 10,000 ants are treated with one application of the insecticide, find the expected value, variance and standard deviation of the number of ants killed.
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