Question

the pto is selling raffle tickets to raise money for classroom supplies. a raffle ticket costs $5. there is 1 winning ticket out of the 300 tickets sold. the winner gets a prize worth $86. round your answers to the nearest cent.
what is the expected value (to you) of one raffle ticket? $\boxed{}$
calculate the expected value (to you) if you purchase 6 raffle tickets. $\boxed{}$
what is the expected value (to the pto) of one raffle ticket? $\boxed{}$
if the pto sells all 300 raffle tickets, how much money can they expect to raise for the classroom supplies? $\boxed{}$

Explanation:

Step1: Calculate the expected value of one raffle ticket

The probability of winning \(P(W)=\frac{1}{300}\), the amount won if you win \(A_W = 86 - 5=81\) (since you paid \(5\) for the ticket), and the probability of losing \(P(L)=\frac{299}{300}\), the amount lost if you lose \(A_L=- 5\).

The formula for expected value \(E(X)=\sum_{i}x_ip_i\). So, \(E(X)=P(W)\times A_W+P(L)\times A_L\)

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Step2: Calculate the expected value of 6 raffle tickets

If \(X_1,X_2,\cdots,X_6\) are the expected - values of each of the 6 tickets, and since \(E(aX)=aE(X)\) (where \(a = 6\) and \(E(X)\) is the expected value of one ticket).

\(E(6X)=6\times(-4.71)=-28.26\)

Step3: Calculate the expected value to the PTO of one raffle ticket

The probability that the PTO loses (i.e., a winner is drawn) \(P(W)=\frac{1}{300}\), the amount the PTO loses if there is a winner \(A_{W - PTO}=-86 + 5=-81\) (they gave out a \(86\) - prize but received \(5\) for the ticket), and the probability that the PTO wins (i.e., no winner is drawn) \(P(L)=\frac{299}{300}\), the amount the PTO wins if there is no winner \(A_{L - PTO}=5\)

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Step4: Calculate the money the PTO can expect to raise

If the expected value of one ticket to the PTO is \(E(X)\approx4.71\) and the number of tickets \(n = 300\)

The total expected amount \(=n\times E(X)\)

\(300\times4.71 = 1413\)

Answer:

• The expected value of one raffle ticket: \(-4.71\)
• The expected value of 6 raffle tickets: \(-28.26\)
• The expected value of one raffle ticket to the PTO: \(4.71\)
• The money the PTO can expect to raise: \(1413\)