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Chapter 4: Problem 11

Five thousand lottery tickets are sold for \(\$ 1\) each. One ticket will win \(\$ 1,000,\) two tickets will win \(\$ 500\) each, and ten tickets will win \(\$ 100\) each. Let \(X\) denote the net gain from the purchase of a randomly selected ticket. a. Construct the probability distribution of \(X\). b. Compute the expected value \(E(X)\) of \(X .\) Interpret its meaning. c. Compute the standard deviation \(\sigma\) of \(X\).

Short Answer

Expert verified
a. Probability distribution: X=999 (1/5000), X=499 (2/5000), X=99 (10/5000), X=-1 (4987/5000). b. E(X) = -0.20, meaning an average loss of $0.20 per ticket. c. Calculate variance then take square root to find \( \sigma \).

Step by step solution

01

Identify Possible Outcomes

The possible net gains from purchasing a ticket are: winning \(1,000 (gain \)999), winning \(500 (gain \)499), winning \(100 (gain \)99), and not winning anything (losing $1). These represent different values of net gain \( X \).
02

Determine Probabilities for Each Outcome

Calculate the probability for each net gain. The total number of tickets is 5,000. The probability of gaining \(999 is \( \frac{1}{5000} \), for \)499 is \( \frac{2}{5000} \), for \(99 is \( \frac{10}{5000} \), and for losing \)1 is \( \frac{4987}{5000} \).
03

Construct the Probability Distribution

Create a table with \( X \) and their corresponding probabilities: - \( X = 999 \) with \( P(X = 999) = \frac{1}{5000} \) - \( X = 499 \) with \( P(X = 499) = \frac{2}{5000} \) - \( X = 99 \) with \( P(X = 99) = \frac{10}{5000} \) - \( X = -1 \) with \( P(X = -1) = \frac{4987}{5000} \).
04

Calculate Expected Value \(E(X)\)

The expected value is found using: \[ E(X) = \sum (x_i \cdot P(x_i)) \] Calculate: \( E(X) = (999 \times \frac{1}{5000}) + (499 \times \frac{2}{5000}) + (99 \times \frac{10}{5000}) + (-1 \times \frac{4987}{5000}) \) Which results in \( E(X) = -0.20 \).
05

Interpretation of Expected Value

An expected value of \( -0.20 \) means that on average, a person loses $0.20 for each ticket purchased.
06

Calculate Standard Deviation \(\sigma\)

First, find the variance using: \[ \text{Var}(X) = \sum ((x_i - E(X))^2 \cdot P(x_i)) \] Substitute values: \( \text{Var}(X) = ((999 - (-0.20))^2 \times \frac{1}{5000}) + ((499 - (-0.20))^2 \times \frac{2}{5000}) + ((99 - (-0.20))^2 \times \frac{10}{5000}) + ((-1 - (-0.20))^2 \times \frac{4987}{5000}) \) Calculate to get the variance, then take the square root to find the standard deviation \( \sigma \).

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Key Concepts

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

Expected Value
When dealing with probability distributions, the expected value is a fundamental concept that helps us find out the average outcome we can anticipate from a random variable. Think of it as the mean or average of a random process.
In the context of lottery tickets, the expected value tells us the average gain or loss per ticket if we repeatedly purchase tickets under the same conditions. It helps answer the question, "In the long run, how much can I expect to win or lose from playing this lottery?"

For our lottery ticket scenario:
  • We calculate this average or expected value by multiplying each possible outcome by its probability and then summing them up.
  • In our case, the outcome of winning with net gains of \(999, \)499, \(99, or losing \)1 (for the non-winning tickets) is weighted by their respective probabilities of occurrence.
  • The formula for expected value is expressed as: \[ E(X) = \sum (x_i \cdot P(x_i)) \]
  • From our calculations, this results in an expected value of \( E(X) = -0.20 \), meaning on average, a person loses $0.20 per ticket bought.
An understanding of expected value helps buyers make informed choices about whether participating in the lottery is a sound financial decision.
Standard Deviation
Standard deviation is a measure that indicates the amount of variability or spread in a set of data. In simpler terms, it tells us how much the outcomes deviate from the average.

In our exercise concerning lottery tickets, the standard deviation helps us understand the level of risk or uncertainty involved with the expected outcome.

Why is it Important?

  • A higher standard deviation means more variability in winnings, suggesting that the actual amount won could be far from the expected value.
  • Conversely, a lower standard deviation indicates that the outcomes are closely packed around the expected value, suggesting more predictability.

Calculation

  • We start by calculating the variance, which is the average of the squared differences from the mean: \[ \text{Var}(X) = \sum ((x_i - E(X))^2 \cdot P(x_i)) \]
  • We then take the square root of the variance to find the standard deviation \( \sigma \).
Understanding standard deviation thus completes our picture of the probability distribution, allowing us to see both the average expected value and the risk associated with the lottery tickets.
Lottery Tickets
Lottery tickets are a popular form of gambling where participants purchase a ticket for a chance to win a prize. The allure of potentially high returns from a low entry cost draws many, despite the uncertainty involved. Understanding the probability distribution, expected value, and standard deviation of such games helps potential buyers make more informed choices.

The Structure of Our Lottery Game

In our scenario, five thousand tickets are sold with several prize categories:
  • 1 ticket at $1,000
  • 2 tickets at $500 each
  • 10 tickets at $100 each
  • 4,987 non-winning tickets

Implications

  • By calculating the expected value, players can discern that they are likely to lose $0.20 on average per ticket.
  • Knowing this loss helps individuals understand the game's risk before choosing to play.
  • By understanding the standard deviation, participants are informed about the variability in their potential outcomes and can account for the risk associated with their investment.
Analyzing lottery tickets in terms of these mathematical concepts underscores the importance of informed decision-making in games of chance.

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Most popular questions from this chapter

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