If a raffle has a winning prize of $100 and each ticket costs $5 with a total of 500 tickets sold, which equation would calculate the expected value?
[tex]$100\left(\frac{1}{500}\right)+(-5)\left(\frac{499}{500}\right)=E(X)$[/tex]
[tex]$95\left(\frac{1}{500}\right)+(-5)\left(\frac{190}{500}\right)=E(X)$[/tex]
[tex]$(100-5)\left(\frac{1}{500}\right)=E(X)$[/tex]
[tex]$(100)\left(\frac{1}{500}\right)=E(X)$[/tex]
Answer (1)
Calculate the net gain if you win: $100 − $5 = $95 .
Determine the probability of winning: 500 1 .
Determine the probability of losing: 500 499 .
Calculate the expected value: E ( X ) = ( $100 ) × ( 500 1 ) + ( − $5 ) × ( 500 499 ) = E ( X ) .
Explanation
Understanding Expected Value Let's break down how to calculate the expected value in this raffle. The expected value represents your average gain or loss if you played the raffle many times. We need to consider the probability of winning and the probability of losing.
Calculating Net Gain and Probability of Winning If you win the raffle, you win $100 , but you also paid $5 for the ticket. So your net gain is $100 − $5 = $95 . The probability of winning is 1 out of 500 tickets, or 500 1 .
Calculating Net Loss and Probability of Losing If you lose the raffle, you only lose the cost of the ticket, which is $5 . So your net loss is $5 . The probability of losing is 499 out of 500 tickets, or 500 499 .
Calculating Expected Value The expected value, E ( X ) , is calculated by multiplying each possible outcome by its probability and then summing those values. In this case:
E ( X ) = ( Net gain if win ) × P ( win ) + ( Net gain if lose ) × P ( lose )
E ( X ) = ( $95 ) × 500 1 + ( − $5 ) × 500 499
Final Answer So the equation that calculates the expected value is:
100 ( 500 1 ) + ( − 5 ) ( 500 499 ) = E ( X )
Examples
Expected value calculations are used in many real-world scenarios, such as insurance, investment decisions, and gambling. For example, insurance companies use expected value to determine how much to charge for premiums. Investors use it to assess the potential profitability of different investments, considering both potential gains and losses. In gambling, understanding expected value can help you determine whether a game is worth playing in the long run. If the expected value is negative, you are likely to lose money over time.
