A community 5K run will award $ 50 to the winner. 40 people enter the race,and they each pay an entry fee of $20. Assuming thay are all equally likely to win, what is a fair price for
the competition? Round to the nearest
cent.
If they are all equally likely to win, the fair price is the expected value of the prize, namely
∑xP(x)
where
x=winning for a particular outcome,
P(x) is the probability of outcome x.
and the summation is summed over all possible outcomes.
The possible outcomes are:
$50 (winner), P($50)=1/40
$0 (others), P(0)=39/40
So the expected value is
∑50*(1/40)+0*(39/40)
=$1.25
To determine a fair price for the competition, we need to consider the total amount collected from entry fees, the number of participants, and the prize money.
The total amount collected from entry fees can be calculated by multiplying the entry fee of $20 by the number of participants, which is 40 in this case. So, the total amount collected is 20 * 40 = $800.
Since there is one winner who will receive a prize of $50, the remaining $750 will be distributed equally among the remaining participants. To find out how much each participant will receive, we need to subtract the prize money from the total amount collected and divide it by the number of participants minus one (since the winner is excluded from the distribution).
Remaining amount = 800 - 50 = $750
Number of participants eligible for distribution = 40 - 1 = 39
Amount each participant will receive = 750 / 39
Now, let's calculate the fair price by adding the amount each participant receives to the entry fee:
Fair price = 20 + (750 / 39)
Calculating this value gives us approximately $38.21 (rounded to the nearest cent). Therefore, a fair price for the competition would be $38.21.