91Ó°ÊÓ

The Tri-State Pick 3 Most states and Canadian provinces have government- sponsored lotteries. Here is a simple lottery wager, from the Tri-State Pick 3 ggame that New Hampshire shares with Maine and Vermont. You choose a number with 3 digits from 0 to \(9 ;\) the state chooses a three-digit winning number at random and pays you \(\$ 500\) if your number is chosen. Because there are 1000 numbers with three digits, you have probability 1\(/ 1000\) of winning. Taking \(X\) to be the amount your ticket pays you, the probability distribution of X is $$\begin{array}{lll}{\text { Payoff } X :} & {\text { \$ 0 }} & {\$ 500} \\\ {\text { Probability: }} & {0.999} & {0.001}\end{array}$$ (a) Show that the mean and standard deviation of \(X\) are \(\mu_{X}=\$ 0.50\) and \(\sigma_{X}=\$ 15.80 .\) (b) If you buy a Pick 3 ticket, your winnings are \(W=X-1,\) bccause it costs \(\$ 1\) to play. Find the mean and standard deviation of W. Interpret each of these values in context.

Step by step solution

01

Finding the Mean of X

The mean of a random variable \(X\), \(\mu_X\), can be found by multiplying each payoff by its probability and summing the results. Thus, \(\mu_X = 0 \times 0.999 + 500 \times 0.001 = 0.50\).

02

Finding the Standard Deviation of X

First, find the expected value of \(X^2\): \(E(X^2) = (0^2 \times 0.999) + (500^2 \times 0.001) = 250\). Then use the formula for the variance: \(\sigma_X^2 = E(X^2) - (\mu_X)^2 = 250 - 0.5^2 = 249.75\). The standard deviation \(\sigma_X\) is the square root of the variance: \(\sigma_X = \sqrt{249.75} \approx 15.80\).

03

Finding the Mean of W

If \(W = X - 1\), to find \(\mu_W\), apply the linear transformation to the mean: \(\mu_W = \mu_X - 1 = 0.50 - 1 = -0.50\).

04

Finding the Standard Deviation of W

For a constant subtraction from a random variable, the standard deviation is unchanged: \(\sigma_W = \sigma_X = 15.80\).

05

Interpret the Mean and Standard Deviation of W

The expected net gain after buying a ticket is \(-\\(0.50\), meaning on average, a player loses 50 cents per ticket bought. The standard deviation of \\)15.80 illustrates the variability in net winnings, showing that payouts (or losses) can fluctuate significantly from the mean.

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Key Concepts

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

Mean

The mean, also known as the expected value, of a random variable is a measure of the central tendency of a probability distribution. It gives us an idea of where the center of the distribution lies.
In the context of the Tri-State Pick 3 lottery, the mean of the random variable \(X\) is calculated as \(\mu_X = 0 \times 0.999 + 500 \times 0.001 = 0.50\). This tells us that, on average, for each ticket you buy, you can expect a payout of \(0.50. However, considering the ticket costs \)1, the expected net gain indicated by \(\mu_W = -0.50\) shows that, on average, you lose 50 cents per ticket.
This kind of mean value is instrumental in understanding potential outcomes over numerous trials, despite any single trial producing a seemingly skewed result.

Standard Deviation

The standard deviation is a measure of the spread or dispersion of a set of values in a probability distribution. It tells us how much the values of a random variable vary from the mean.
For the lottery, the standard deviation of \(X\) is calculated by first finding the variance \(\sigma_X^2 = E(X^2) - (\mu_X)^2 = 249.75\) and then taking the square root, resulting in \(\sigma_X \approx 15.80\).
This standard deviation indicates that there is a significant spread around the mean value. It shows that the outcomes can greatly differ from the mean payout of $0.50. When it comes to evaluating \(W = X - 1\), the standard deviation remains unchanged at 15.80. This portrays the potential volatility and variability in net winnings for each ticket purchase.

Linear Transformation

A linear transformation involves changing a random variable by adding or multiplying by a constant. It directly affects the mean but typically does not affect the standard deviation, except when multiplying by a constant.
In our example, the winnings \(W = X - 1\) demonstrate a linear transformation where $1 is subtracted from the random variable \(X\). For the mean, \(\mu_W = \mu_X - 1 = 0.50 - 1 = -0.50\), illustrating an expected loss per lottery ticket.
However, the standard deviation \(\sigma_W\) remains the same as \(\sigma_X\), emphasizing that changing the location of the distribution does not alter its spread. This property is crucial for discharging the effects of fixed costs, such as the ticket price, in real-world scenarios.

Random Variable

A random variable is a fundamental concept in probability representing the outcomes of a random phenomenon. It assigns numerical values to each possible outcome in a probability space.
In the lottery context, the random variable \(X\) represents the payout of the ticket, with possible values of \(0 or \)500.
The probability distribution of \(X\) tells us that the chance of winning $500 is 0.001, while the probability of winning nothing is 0.999. When transformed into \(W = X - 1\), the random variable now reflects the net winnings or losses, showcasing an expected negative financial outcome. Understanding random variables enables one to forecast and interpret the likelihood of various outcomes, providing a mathematical basis for decision-making under uncertainty.