Expected Value in North Carolina’s Pick 4 Game In North Carolina’s Pick 4 lottery game, you can pay $1 to select a four-digit number from 0000 through 9999. If you select the same sequence of four digits that are drawn, you win and collect $5000.

d. Find the expected value.

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Step 1: Understand the problem. The expected value (E[X]) is a measure of the average outcome of a random event over the long run. In this case, we are calculating the expected value of playing the Pick 4 lottery game, where you pay $1 to play and can win $5000 if your chosen number matches the winning number.

Step 2: Identify the probabilities. There are 10,000 possible four-digit combinations (from 0000 to 9999). The probability of selecting the correct number is $ P(\text{win}) = \frac{1}{10000} $, and the probability of not winning is $ P(\text{lose}) = 1 - \frac{1}{10000} = \frac{9999}{10000} $.

Step 3: Define the outcomes. If you win, your net gain is $ \$5000 - \$1 = \$4999 $. If you lose, your net gain is $ -\$1 $ (since you lose the $1 you paid to play).

Step 4: Use the expected value formula. The formula for expected value is $ E[X] = \sum (x_i \cdot P(x_i)) $, where $ x_i $ represents each possible outcome and $ P(x_i) $ is the probability of that outcome. Substitute the values: $ E[X] = (4999 \cdot \frac{1}{10000}) + (-1 \cdot \frac{9999}{10000}) $.

Step 5: Simplify the expression. Combine the terms to calculate the expected value: $ E[X] = \frac{4999}{10000} - \frac{9999}{10000} $. This will give you the expected value of playing the game in dollars.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Expected Value

Expected value is a fundamental concept in probability and statistics that represents the average outcome of a random event when repeated many times. It is calculated by multiplying each possible outcome by its probability and summing these products. In the context of a lottery game, the expected value helps determine the average amount a player can expect to win or lose per game played.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In the Pick 4 lottery, the probability of selecting the winning four-digit number is 1 in 10,000, since there are 10,000 possible combinations (from 0000 to 9999). Understanding probability is essential for calculating the expected value and assessing the game's risk.

Payoff

Payoff refers to the amount of money won or lost in a gambling scenario, which is crucial for calculating expected value. In the Pick 4 game, if a player wins, they receive a payoff of $5,000 for their $1 bet. The payoff, combined with the probability of winning, directly influences the expected value calculation, helping players understand the potential financial outcome of their bets.

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