1000 tickets for prizes are sold for 42 each. Seven prizes will be awarded-one for $400, one for $200, and five for 450. steven purchases one of the tickets. find the expected value and find the fair price of the ticket.
The expected value of one ticket is the total prize money divided by 1000. That is just $5.25. The price Steve paid is much too high. Whatever is a "fair" price is a matter of opinion, or how charitable the ticket buyer wants to be.
To find the expected value, you need to calculate the probability of winning each prize and multiply it by the corresponding prize amount.
Let's break down the problem step by step:
Step 1: Find the total amount collected from ticket sales.
Since there are 1000 tickets sold for $42 each, the total amount collected is $42 * 1000 = $42,000.
Step 2: Calculate the probability of winning each prize.
There are a total of 7 prizes:
- 1 prize for $400
- 1 prize for $200
- 5 prizes for $50
So, the probability of winning the $400 prize is 1/1000 (as there is only one $400 prize out of 1000 tickets).
The probability of winning the $200 prize is also 1/1000.
The probability of winning each $50 prize is 5/1000, as there are five such prizes out of the 1000 tickets.
Step 3: Calculate the expected value.
The expected value is the sum of the products of the probability of winning each prize and the corresponding prize amount.
Expected value = (Probability of $400 prize * $400) + (Probability of $200 prize * $200) + (Probability of each $50 prize * $50)
Expected value = (1/1000 * $400) + (1/1000 * $200) + (5/1000 * $50) = $0.40 + $0.20 + $0.25 = $0.85
Therefore, the expected value for Steven purchasing a ticket is $0.85.
Step 4: Calculate the fair price of the ticket.
The fair price of the ticket is the amount that ensures no profit or loss for the player in the long run. This means that the expected value should be equal to the cost of the ticket.
So, the fair price of the ticket is $0.85.
Please note that in this case, the expected value is less than the ticket price, indicating a slight expected loss for the player on average.