To determine which equation correctly depicts the calculation of the expected value for a ticket, we need to consider the possible outcomes and their associated probabilities and values.
1. Winning the Car:
- Value: [tex]$30,000
- Probability: \(\frac{1}{5000}\)
- Expected Value Contribution: \(30,000 \times \frac{1}{5000}\)
2. Winning a Gift Card:
- Value: $[/tex]100
- Probability: [tex]\(\frac{5}{5000}\)[/tex] (because there are 5 gift cards and 5000 total tickets)
- Expected Value Contribution: [tex]\(100 \times \frac{5}{5000}\)[/tex]
3. Losing (Not winning a car or a gift card):
- Value: $-20 (cost of the ticket)
- Probability: [tex]\(\frac{4994}{5000}\)[/tex] (since [tex]\(\frac{1}{5000}\)[/tex] + [tex]\(\frac{5}{5000}\)[/tex] are the probabilities of winning, so losing probability is [tex]\(1 - \left(\frac{1}{5000} + \frac{5}{5000}\right) = \frac{4994}{5000}\)[/tex])
- Expected Value Contribution: [tex]\(-20 \times \frac{4994}{5000}\)[/tex]
Now we sum these contributions to find the expected value [tex]\(E(X)\)[/tex]:
[tex]\[
E(X) = 30,000 \left(\frac{1}{5000}\right) + 100 \left(\frac{5}{5000}\right) + (-20) \left(\frac{4994}{5000}\right)
\][/tex]
Breaking down the calculation:
- [tex]\(30,000 \left(\frac{1}{5000}\right) = 6.0\)[/tex]
- [tex]\(100 \left(\frac{5}{5000}\right) = 0.1\)[/tex]
- [tex]\(-20 \left(\frac{4994}{5000}\right) = -19.976\)[/tex]
Adding these components, we get:
[tex]\[
E(X) = 6.0 + 0.1 + (-19.976) = -13.876
\][/tex]
The correct equation form which describes this is:
[tex]\[
30,000\left(\frac{1}{5000}\right)+100\left(\frac{1}{1000}\right)+(-20)\left(\frac{4994}{5000}\right)=E(X)
\][/tex]
However, simplifying the fractions in the last term:
[tex]\[
\frac{4994}{5000} = \left(\frac{5000 - 1 - 5}{5000}\right) = \frac{4994}{5000}
\][/tex]
We have:
[tex]\[
(-20) \left(\frac{5000 - 1 - 5}{5000}\right)
\][/tex]
So, the complete equation reflecting this scenario is:
[tex]\[
30,000\left(\frac{1}{5000}\right)+100\left(\frac{1}{1000}\right)+(-20)\left(\frac{5000 - 1 - 5}{5000}\right)=E(X)
\][/tex]
Thus, the correct answer is:
[tex]\[30,000\left(\frac{1}{5000}\right)+100\left(\frac{1}{1000}\right)+(-20)\left(\frac{5000 - 1 - 5}{5000}\right)=E(X)\][/tex]
