Question

the student council is hosting a drawing to raise money for scholarships. they are selling tickets for $8 each and will sell 800 tickets. there is one $2,000 grand prize, four $400 second prizes, and twelve $30 third prizes. you just bought a ticket. find the expected value for your profit. round to the nearest cent. $

Explanation:

Step1: Calculate probabilities

The probability of winning the grand - prize: $P_1=\frac{1}{800}$. The probability of winning a second - prize: $P_2 = \frac{4}{800}=\frac{1}{200}$. The probability of winning a third - prize: $P_3=\frac{12}{800}=\frac{3}{200}$. The probability of winning nothing: $P_4 = 1-( \frac{1}{800}+\frac{4}{800}+\frac{12}{800})=\frac{800-(1 + 4+12)}{800}=\frac{783}{800}$.

Step2: Calculate profits for each case

If you win the grand - prize, your profit is $2000 - 8=1992$. If you win a second - prize, your profit is $400 - 8 = 392$. If you win a third - prize, your profit is $30 - 8=22$. If you win nothing, your profit is $- 8$.

Step3: Calculate expected value

The expected value $E$ of a discrete random variable is given by $E=\sum_{i}x_iP_i$. So $E=(1992)\times\frac{1}{800}+(392)\times\frac{1}{200}+(22)\times\frac{3}{200}+(-8)\times\frac{783}{800}$.
First term: $(1992)\times\frac{1}{800}=\frac{1992}{800}=2.49$.
Second term: $(392)\times\frac{1}{200}=\frac{392}{200}=1.96$.
Third term: $(22)\times\frac{3}{200}=\frac{66}{200}=0.33$.
Fourth term: $(-8)\times\frac{783}{800}=-\frac{6264}{800}=- 7.83$.
Then $E=2.49 + 1.96+0.33-7.83=-3.05$.

Answer:

$-3.05$