Question

a school fundraiser sells 1,200 raffle tickets. each ticket costs $2. there is one grand prize worth $100 and 5 smaller prizes worth $20 each. what is the expected gain or loss for buying 1 ticket?

Explanation:

Step1: Calculate probability of winning grand - prize

The probability of winning the grand - prize $P(G)$ is $\frac{1}{1200}$ since there is 1 grand - prize out of 1200 tickets. The gain from winning the grand - prize is $100 - 2=98$ (subtracting the cost of the ticket).

Step2: Calculate probability of winning a small prize

The probability of winning a small prize $P(S)$ is $\frac{5}{1200}=\frac{1}{240}$. The gain from winning a small prize is $20 - 2 = 18$.

Step3: Calculate probability of winning no prize

The probability of winning no prize $P(N)$ is $1-\frac{1 + 5}{1200}=1-\frac{6}{1200}=\frac{1194}{1200}$. The gain from winning no prize is $- 2$ (the cost of the ticket).

Step4: Calculate the expected value

The expected value $E$ is given by the formula $E=P(G)\times98+P(S)\times18+P(N)\times(-2)$.
\[

$$\begin{align*} E&=\frac{1}{1200}\times98+\frac{1}{240}\times18+\frac{1194}{1200}\times(-2)\\ &=\frac{98}{1200}+\frac{18}{240}-\frac{2388}{1200}\\ &=\frac{98 + 90-2388}{1200}\\ &=\frac{188 - 2388}{1200}\\ &=\frac{-2200}{1200}\approx - 1.83 \end{align*}$$

\]

Answer:

loss of $1.83$