200 tickets are sold to a raffle. It costs $5 to buy a ticket. There is one first prize winner and one second prize winner. First prize is $500. Second prize is $250. What is the value of playing this game?
To determine the value of playing this game, we need to calculate the expected value. The expected value is the sum of all possible outcomes multiplied by their respective probabilities.
First, let's calculate the total revenue from selling the tickets. Since there are 200 tickets sold at $5 each, the total revenue is given by:
Total Revenue = Number of Tickets × Cost per Ticket
Total Revenue = 200 × $5
Total Revenue = $1000
Next, let's calculate the probability of winning each prize. There is only one first prize winner out of 200 tickets, so the probability of winning the first prize is given by:
Probability of Winning First Prize = 1 / Number of Tickets
Probability of Winning First Prize = 1 / 200
Probability of Winning First Prize = 0.005
Similarly, there is only one second prize winner out of 200 tickets, so the probability of winning the second prize is also 0.005.
Now, let's calculate the expected value of each prize. The expected value of the first prize is the prize amount multiplied by the probability of winning:
Expected Value of First Prize = Prize Amount × Probability of Winning First Prize
Expected Value of First Prize = $500 × 0.005
Expected Value of First Prize = $2.50
Similarly, the expected value of the second prize is:
Expected Value of Second Prize = Prize Amount × Probability of Winning Second Prize
Expected Value of Second Prize = $250 × 0.005
Expected Value of Second Prize = $1.25
Finally, the total expected value is the sum of the expected values of all prizes:
Total Expected Value = Expected Value of First Prize + Expected Value of Second Prize
Total Expected Value = $2.50 + $1.25
Total Expected Value = $3.75
Therefore, the value of playing this game is $3.75, which means on average, a player can expect to win $3.75.