a charity is holding a raffle to raise money. there is one car worth $30,000 and five $100 gift - cards…

a charity is holding a raffle to raise money. there is one car worth $30,000 and five $100 gift - cards being raffled off. each ticket costs $20, and there are a total of 5,000 tickets being sold. which equation correctly depicts the calculation of the expected value for a ticket?\n$30,000\\left(\\frac{1}{5000}\\right)+100\\left(\\frac{1}{1000}\\right)+(-20)\\left(\\frac{2497}{2500}\\right)=e(x)$\n$29,980\\left(\\frac{1}{5000}\\right)+80\\left(\\frac{1}{1000}\\right)+(-20)\\left(\\frac{2497}{2500}\\right)=e(x)$\n$30,000\\left(\\frac{1}{5000}\\right)+100\\left(\\frac{1}{1000}\\right)=e(x)$\n$29,980\\left(\\frac{1}{5000}\\right)+80\\left(\\frac{1}{1000}\\right)=e(x)$

a charity is holding a raffle to raise money. there is one car worth $30,000 and five $100 gift - cards being raffled off. each ticket costs $20, and there are a total of 5,000 tickets being sold. which equation correctly depicts the calculation of the expected value for a ticket?\n$30,000\\left(\\frac{1}{5000}\\right)+100\\left(\\frac{1}{1000}\\right)+(-20)\\left(\\frac{2497}{2500}\\right)=e(x)$\n$29,980\\left(\\frac{1}{5000}\\right)+80\\left(\\frac{1}{1000}\\right)+(-20)\\left(\\frac{2497}{2500}\\right)=e(x)$\n$30,000\\left(\\frac{1}{5000}\\right)+100\\left(\\frac{1}{1000}\\right)=e(x)$\n$29,980\\left(\\frac{1}{5000}\\right)+80\\left(\\frac{1}{1000}\\right)=e(x)$

Answer

Explanation:

Step1: Calculate probabilities

The probability of winning the car is $\frac{1}{5000}$ since there is 1 car and 5000 tickets. The probability of winning a $100 gift - card is $\frac{5}{5000}=\frac{1}{1000}$ as there are 5 gift - cards and 5000 tickets. The probability of winning nothing is $1-\frac{1 + 5}{5000}=1-\frac{6}{5000}=\frac{4994}{5000}=\frac{2497}{2500}$.

Step2: Calculate net - gains

If you win the car worth $30000$, but you paid $20$ for the ticket, the net - gain is $30000 - 20=29980$. If you win a $100$ gift - card and paid $20$ for the ticket, the net - gain is $100 - 20 = 80$. If you win nothing, the net - gain is $- 20$.

Step3: Use expected - value formula

The expected - value formula is $E(X)=\sum_{i}x_ip_i$, where $x_i$ are the possible values and $p_i$ are their corresponding probabilities. So $E(X)=29980\left(\frac{1}{5000}\right)+80\left(\frac{1}{1000}\right)+(-20)\left(\frac{2497}{2500}\right)$.

Answer:

$29980\left(\frac{1}{5000}\right)+80\left(\frac{1}{1000}\right)+(-20)\left(\frac{2497}{2500}\right)=E(X)$