Use the following statement to answer parts a) and b). One hundred raffle tickets are sold for $3 each. One prize is to be awarded. Raul purchases one ticket. a). Determine his expected value. b). The fair price of a ticket is?
$300 is what they will take in.
How much is the prize? I need that to find expected value for Raul.
The fair price of a ticket depends on the pay out.
To answer parts a) and b), we need to calculate the expected value of Raul's ticket purchase.
a) The expected value is calculated by multiplying each possible outcome by its probability and summing them up.
In this case, there are two possible outcomes: Raul wins the prize (probability of winning is 1/100) or Raul does not win the prize (probability of not winning is 99/100).
If Raul wins the prize, he will receive the value of the prize, which we don't have information about. Let's represent it as "x" dollars.
If Raul does not win the prize, he will not receive any money.
Therefore, the expected value (EV) can be calculated as:
EV = (Probability of winning * Value if winning) + (Probability of not winning * Value if not winning)
EV = (1/100 * x) + (99/100 * 0)
EV = x/100
Since we don't have information about the value of the prize, we cannot determine the exact expected value for Raul.
b) The fair price of a ticket is the amount that would make the expected value zero, indicating a fair game. In other words, it is the price at which Raul would be neither gaining nor losing money, on average.
To find the fair price, we set the EV to zero:
0 = x/100
This equation tells us that the value of the prize (x) would need to be zero for the expected value to be zero. However, this scenario doesn't make sense for a raffle prize. Typically, the value of the prize would be higher than zero.
Therefore, since we don't have information for the value of the prize, we cannot determine the fair price of a ticket.
To answer parts (a) and (b) of the question, we need to understand the concept of expected value.
The expected value is a measure that tells us the average outcome we can expect from a random event. To calculate the expected value, we multiply each possible outcome by its probability and sum the results.
Given the information provided, let's calculate the expected value and the fair price of a ticket.
a) To determine Raul's expected value, we need to know the probability of winning the prize. Since there are 100 raffle tickets sold and only one prize, Raul's probability of winning is 1/100, or 0.01.
Now, let's calculate the expected value using the formula:
Expected Value = Value of Winning * Probability of Winning
Value of Winning: Raul will receive the prize if he wins, which can be considered as $0 (since there is no information about the value of the prize).
Expected Value = $0 * 0.01 = $0
Therefore, Raul's expected value is $0.
b) The fair price of a ticket is the amount at which the expected value is zero. In other words, if the price of a ticket is set at the fair price, the expected winnings for each participant would be balanced.
Since Raul's expected value is $0, the fair price of a ticket would be the price at which the total expected winnings from selling 100 tickets is equal to the cost of selling those tickets.
To determine this, we multiply the expected value of each ticket ($0) by the number of tickets sold (100) and set it equal to the total cost of selling the tickets ($3 per ticket * 100 tickets = $300).
0 * 100 = $3 * 100
0 = $300
Since the equation is not balanced, the fair price of a ticket is not $3. In fact, since Raul's expected value is $0, the fair price should be lower than $3 to make the equation balanced. Unfortunately, we cannot determine the exact fair price without knowing the value of the prize.
In summary:
a) Raul's expected value is $0.
b) The fair price of a ticket is unknown without additional information about the value of the prize.