Question

chapter 4: discrete probability and the binomial distribution
score: 5/18 answered: 3/8
question 4
the student council is hosting a drawing to raise money for scholarships. they are selling tickets for $8 each and will sell 700 tickets. there is one $3,000 grand prize, three $400 second prizes, and thirteen $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 of winning each prize

The probability of winning the grand - prize ($P_1$) is $\frac{1}{700}$ since there is 1 grand - prize and 700 tickets. The probability of winning a second - prize ($P_2$) is $\frac{3}{700}$ as there are 3 second - prizes. The probability of winning a third - prize ($P_3$) is $\frac{13}{700}$ as there are 13 third - prizes. The probability of winning nothing ($P_0$) is $1-\frac{1 + 3+13}{700}=1-\frac{17}{700}=\frac{683}{700}$.

Step2: Calculate the net gain for each outcome

The net gain if you win the grand - prize is $3000 - 8=2992$. The net gain if you win a second - prize is $400 - 8 = 392$. The net gain if you win a third - prize is $10 - 8=2$. The net gain if you win nothing is $- 8$.

Step3: Calculate the expected value

The expected value $E$ is given by the formula $E=P_1\times2992+P_2\times392+P_3\times2+P_0\times(-8)$.
\[

$$\begin{align*} E&=\frac{1}{700}\times2992+\frac{3}{700}\times392+\frac{13}{700}\times2+\frac{683}{700}\times(-8)\\ &=\frac{2992+3\times392 + 13\times2-683\times8}{700}\\ &=\frac{2992 + 1176+26-5464}{700}\\ &=\frac{4194 - 5464}{700}\\ &=\frac{-1270}{700}\approx - 1.81 \end{align*}$$

\]

Answer:

$-1.81$