Questions: Lottery: In the New York State Numbers Lottery, you pay 1 and pick a number from 000 to 999. If your number comes up, you win 550, which is a profit of 549. If you lose, you lose 1. Your probability of winning is 0.001. Part 1 of 2 (a) What is the expected value of your profit? Round the answer to two decimal places. The expected value of profit is -.45. Alternate Answer: The expected value of profit is -0.45. Part: 1 / 2 Part 2 of 2 (b) Is it an expected gain or an expected loss? Round the answer to two decimal places.

Solution Steps

To solve this problem, we need to calculate the expected value of the profit from playing the lottery. The expected value is calculated by multiplying each possible outcome by its probability and summing these products. In this case, there are two outcomes: winning and losing. The probability of winning is given, and the probability of losing can be derived from it. We then calculate the expected value using these probabilities and the respective profits or losses.

We are given a lottery scenario where:

• The cost to play is \$1.
• The prize for winning is \$550, resulting in a profit of \$549.
• The probability of winning is 0.001.
• The probability of losing is \(1 - 0.001 = 0.999\).

We need to calculate the expected value of the profit and determine if it is an expected gain or loss.

The expected value \(E\) of a random variable is calculated using the formula:

\[ E = \sum ( \text{Probability of outcome} \times \text{Value of outcome} ) \]

In this case, we have two outcomes: winning and losing.

• Winning:

  • Probability = 0.001
  • Profit = \$549

• Losing:

  • Probability = 0.999
  • Loss = -\$1

The expected value \(E\) is:

\[ E = (0.001 \times 549) + (0.999 \times (-1)) \]

Calculating each term:

\[ 0.001 \times 549 = 0.549 \]

\[ 0.999 \times (-1) = -0.999 \]

Adding these results:

\[ E = 0.549 - 0.999 = -0.450 \]

Since the expected value is negative, it indicates an expected loss.

Final Answer