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.

1. With a single $1 bet, what is the probability that you win $500?

2. Let X denote your winnings for a $1 bet, so x = $0 or x = $500.

3. Construct the probability distribution for X.

4. 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.

5. In Ohio’s Pick 4 lottery, you pick one of the 10,000 four-digit 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 o -playing Pick 4, or playing Pick 3 in which you win $500 for a correct choice of a three-digit number? Justify your answer.

1. With a single $1 bet, the probability that you win $500 is 1/1000, or 0.001.

2. The probability distribution for X is as follows:
X = 0, P(X) = 999/1000
X = 500, P(X) = 1/1000

3. The mean of the distribution equals 0.50, corresponding to an expected return of 50 cents for the dollar paid to play. This means that, on average, you can expect to get back 50 cents for every dollar you spend on the lottery.

4. In terms of expected winnings, playing Pick 4 is better than playing Pick 3, since you can win $5000 with a correct choice of a four-digit number, compared to only $500 with a correct choice of a three-digit number.

1. The probability of winning $500 with a single $1 bet can be calculated by dividing the number of favorable outcomes (i.e., winning numbers) by the total number of possible outcomes. In this case, there is only 1 winning number out of a total of 1000 possible numbers, so the probability of winning $500 is 1/1000.

2. In this scenario, X represents the random variable representing your winnings. It can take the values x = $0 or x = $500.

3. To construct the probability distribution for X, we assign probabilities to each possible outcome. Since there is only 1 winning number out of 1000, the probability of winning $500 is 1/1000, while the probability of winning $0 is 999/1000 (since there are 999 losing numbers). Therefore, the probability distribution for X is as follows:

X = $0 with probability 999/1000
X = $500 with probability 1/1000

4. To calculate the mean of the distribution, we multiply each possible outcome by its corresponding probability and sum them up. In this case, the mean (expected value) can be calculated as:

Mean = ($0 * 999/1000) + ($500 * 1/1000) = $0 + $0.50 = $0.50

The mean of the distribution is $0.50, which corresponds to an expected return of 50 cents for each dollar paid to play. This means that, on average, you can expect to lose 50 cents for every dollar you spend on the Pick 3 game.

5. To determine whether playing Pick 4 or Pick 3 is better in terms of expected winnings, we compare the expected values of the two games.

In Pick 4, the probability of winning $5000 is 1/10000 (since there is only 1 winning number out of 10000), while the probability of winning $0 is 9999/10000. The expected value for Pick 4 can be calculated as:

Expected value (Pick 4) = ($0 * 9999/10000) + ($5000 * 1/10000) = $0 + $0.50 = $0.50

Comparing this to the expected value of $0.50 (50 cents) for the Pick 3 game, it is clear that the expected winnings are the same for both games. Therefore, in terms of expected winnings, it does not matter whether you play Pick 3 or Pick 4.

1. To calculate the probability of winning $500 with a single $1 bet in the Pick 3 game, we need to determine the total number of possible outcomes and the favorable outcomes.

Total number of possible outcomes: There are 1000 three-digit numbers between 000 and 999.

Favorable outcomes: Since you only win $500 if your chosen number is selected, there is only one favorable outcome.

Therefore, the probability of winning $500 with a single $1 bet is 1/1000.

2. To define the random variable X, we assign the value $0 if you don't win and $500 if you win. So, X = {$0, $500}.

3. To construct the probability distribution for X, we need to assign probabilities to each possible outcome.

For X = $0: The probability of not winning is 999/1000 since there are 999 unfavorable outcomes out of a total of 1000.

For X = $500: The probability of winning is 1/1000, as calculated in question 1.

Therefore, the probability distribution for X is:

X | $0 | $500
---------------------------------
P(X) | 999/1000 | 1/1000

4. To show that the mean of the distribution equals 0.50, we calculate the expected value (mean) using the probability distribution.

Mean (expected value) = (X1 * P(X1)) + (X2 * P(X2)) + ... + (Xn * P(Xn))

Mean = ($0 * 999/1000) + ($500 * 1/1000)
= $0 + $0.50
= $0.50

The mean of the distribution corresponds to an expected return of 50 cents for the dollar paid to play. This means that, on average, you can expect to win 50 cents for every dollar you spend playing the Pick 3 game.

5. To compare the expected winnings between the Pick 3 and Pick 4 games, we need to calculate the expected values for each game.

In the Pick 4 game, the probability of winning $5000 with a $1 bet is 1/10000, as there are 10000 possible four-digit numbers.

Expected winnings in Pick 4 = ($0 * 9999/10000) + ($5000 * 1/10000)
= $0 + $0.50
= $0.50

Comparing the expected winnings, we find that both Pick 3 and Pick 4 have the same expected value of $0.50. Therefore, in terms of expected winnings, you are better off playing either game. However, it's important to note that winning probabilities and prize amounts are different, so personal preferences and risk tolerance may also play a role in choosing the game.