Question

the student council is hosting a drawing to raise money for scholarships. they are selling tickets for $10 each and will sell 900 tickets. there is one $1,000 grand prize, two $400 second prizes, and ten $10 third prizes. you just bought a ticket. find the expected value for your profit. round to the nearest cent.
$
hint:
hint
video on expected value +

Explanation:

Step1: Calculate probabilities

The probability of winning the grand - prize $P_1=\frac{1}{900}$, the probability of winning a second - prize $P_2 = \frac{2}{900}$, the probability of winning a third - prize $P_3=\frac{10}{900}$, and the probability of winning nothing $P_4=\frac{900-(1 + 2+10)}{900}=\frac{887}{900}$.

Step2: Calculate profit for each case

The cost of a ticket is $C = 10$. If you win the grand - prize, your profit $X_1=1000 - 10=990$. If you win a second - prize, your profit $X_2 = 400-10 = 390$. If you win a third - prize, your profit $X_3=10 - 10=0$. If you win nothing, your profit $X_4=-10$.

Step3: Calculate expected value

The formula for the expected value $E(X)$ is $E(X)=\sum_{i = 1}^{n}x_ip_i$.
\[

$$\begin{align*} E(X)&=X_1P_1+X_2P_2+X_3P_3+X_4P_4\\ &=990\times\frac{1}{900}+390\times\frac{2}{900}+0\times\frac{10}{900}+(- 10)\times\frac{887}{900}\\ &=\frac{990 + 390\times2+0-10\times887}{900}\\ &=\frac{990+780 - 8870}{900}\\ &=\frac{1770 - 8870}{900}\\ &=\frac{-7100}{900}\approx - 7.89 \end{align*}$$

\]

Answer:

$-7.89$