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Playing the lottery The state of Ohio has several statewide lottery options. One is the Pick 3 game in which you pick one of the 1000 three-digit numbers between 000 and 999\. The lottery selects a three-digit number at random. With a bet of \(\$ 1,\) you win \(\$ 500\) if your number is selected and nothing (\$0) otherwise. (Many states have a very similar type of lottery.) (Source: Background information from www.ohiolottery.com.) a. With a single \(\$ 1\) bet, what is the probability that you win \(\$ 500 ?\) b. Let \(X\) denote your winnings for a \(\$ 1\) bet, so \(x=\$ 0\) or \(x=\$ 500\). Construct the probability distribution for \(X\). c. Show that the mean of the distribution equals 0.50 , corresponding to an expected return of 50 cents for the dollar paid to play. Interpret the mean. d. In Ohio's Pick 4 lottery, you pick one of the 10,000 fourdigit numbers between 0000 and 9999 and (with a \(\$ 1\) bet) win \(\$ 5000\) if you get it correct. In terms of your expected winnings, with which game are you better off \(-\) playing Pick \(4,\) or playing Pick 3 ? Justify your answer.

Step by step solution

01

Calculate the Probability of Winning Pick 3

In the Pick 3 game, you need to select the correct three-digit number from 000 to 999. This means there are 1000 possible outcomes. Since only one of these outcomes will result in a win, the probability of winning is the reciprocal of 1000. The probability, \( P( ext{win Pick 3}) = \frac{1}{1000} = 0.001 \).

02

Construct the Probability Distribution for X in Pick 3

The variable \(X\) represents your winnings. You win \(\\(500\) with a probability of \(0.001\), and you win \(\\)0\) with a probability of \(1 - 0.001 = 0.999\). Thus, the probability distribution for \(X\) is:\[P(X = \\(500) = 0.001 \P(X = \\)0) = 0.999.\]

03

Calculate the Mean of the Distribution for Pick 3

The mean or expected value \(E(X)\) is calculated as the sum of each outcome multiplied by its probability.\[E(X) = 500 \times 0.001 + 0 \times 0.999 = 0.5.\]This means the expected return is \\(0.50 for every dollar paid, implying an average loss of \\)0.50.

04

Calculate the Probability of Winning Pick 4

In the Pick 4 game, you select one correct four-digit number from 0000 to 9999, thus having 10,000 possible outcomes. The probability of winning is the reciprocal of 10,000. The probability, \( P( ext{win Pick 4}) = \frac{1}{10000} = 0.0001 \).

05

Construct the Probability Distribution for X in Pick 4

For Pick 4, the variable \(X\) represents winnings of \(\\(5000\) with a probability of \(0.0001\) and \(\\)0\) with a probability of \(0.9999\).Thus, the probability distribution for \(X\) is:\[P(X = \\(5000) = 0.0001 \P(X = \\)0) = 0.9999.\]

06

Calculate the Mean of the Distribution for Pick 4

The mean or expected value \(E(X)\) for Pick 4 is:\[E(X) = 5000 \times 0.0001 + 0 \times 0.9999 = 0.5.\]This means the expected return in Pick 4 is \$0.50 for every dollar paid.

07

Compare Expected Winnings for Pick 3 and Pick 4

Both Pick 3 and Pick 4 have an expected value of \\(0.50 winnings per \\)1 bet, suggesting equivalent expected losses in terms of expected monetary return.The answer doesn't advocate for a preference between games based purely on this analysis.

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Key Concepts

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

Probability Distribution

A probability distribution represents how probabilities are distributed over various possible outcomes of a random event. In the context of the Ohio Pick 3 lottery, we have two possible outcomes when you place a \(1 bet:

• You win \)500 if your number matches the drawn number.
• You win nothing (\(0) if it doesn't.

The probability distribution is constructed by pairing each outcome with its probability. In this case, the probabilities are:

• The probability of winning \)500 is \( P(X = 500) = 0.001 \).
• The probability of winning $0 is \( P(X = 0) = 0.999 \).

This distribution gives a complete picture of what happens in the lottery draw when you place a bet.

Expected Value

The expected value is a crucial concept in probability that helps determine the average outcome of a random event based on its probability distribution. For the Pick 3 lottery, the expected value can be thought of as the average return per game played over a large number of repetitions.
To calculate the expected value, we multiply each possible outcome by its probability and sum up the results:\[E(X) = 500 \times 0.001 + 0 \times 0.999 = 0.5\]This result, \(0.50, represents the expected earnings for each \)1 lottery ticket purchased. It effectively predicts that over many games, on average, you will receive 50 cents back for each dollar you spend.

Lottery Odds

Lottery odds refer to the probability of winning a particular lottery game. In Pick 3, you choose a specific three-digit number among a possible 1000 combinations. Hence, your probability of winning—or the odds—is simply:\[P(\text{win Pick 3}) = \frac{1}{1000} = 0.001\]Therefore, the odds are 1 in 1000.

• This very low probability highlights the challenge of winning the lottery.

For the Pick 4 lottery, where you choose one number out of 10,000, the probability of winning shrinks further to:\[P(\text{win Pick 4}) = \frac{1}{10000} = 0.0001\]The smaller probability in Pick 4 compared to Pick 3 suggests even steeper odds, making it harder to win.

Mean of Distribution

The mean of a probability distribution provides insight into the expected average outcome of an event. For the Pick 3 lottery, the mean of the distribution reflects the average winnings over time.
In calculating the mean, we found that it equals 0.50, which indicates the average amount expected per $1 bet:

• \(E(X) = 500 \times 0.001 + 0 \times 0.999 = 0.5\)

This result implies a typical loss of 50 cents per dollar invested in this game, reinforcing the idea that while a big prize is possible, playing the game consistently results in a loss on average.