Question

a cash prize of $2400 is to be awarded at a fundraiser. if 1200 tickets are sold at $7 each, find the expected value. round your answer to the nearest cent, if necessary. the expected value is dollar(s).

Explanation:

Step1: Calculate probability of winning

The probability $P(\text{win})$ of winning is the number of winning - tickets (1) divided by the total number of tickets sold. There is 1 winning - ticket and 1200 tickets sold, so $P(\text{win})=\frac{1}{1200}$. The probability of losing $P(\text{lose}) = 1 - P(\text{win})=1-\frac{1}{1200}=\frac{1199}{1200}$.

Step2: Determine the net gain for winning and losing

If you win, the net gain $X_1$ is the prize money minus the cost of the ticket. The prize money is $2400$ and the ticket cost is $7$, so $X_1 = 2400 - 7=2393$. If you lose, the net gain $X_2$ is just the cost of the ticket (a loss), so $X_2=-7$.

Step3: Calculate the expected value

The formula for the expected value $E(X)$ of a discrete - random variable is $E(X)=P(\text{win})\times X_1+P(\text{lose})\times X_2$. Substitute the values: $E(X)=\frac{1}{1200}\times2393+\frac{1199}{1200}\times(-7)$. First, calculate $\frac{1}{1200}\times2393=\frac{2393}{1200}\approx1.9942$ and $\frac{1199}{1200}\times(-7)=\frac{-8393}{1200}\approx - 6.9942$. Then $E(X)=\frac{2393 - 8393}{1200}=\frac{-6000}{1200}=-5$.

Answer:

$-5$