Score: 0 of 4 points You buy a lottery ticket to a lottery that costs $$ \$ 15 $$ per ticket. There are only 300 tickets available to be sold in this lottery. In this lottery there are one $$ \$ 500 $$ prize(s), two $$ \$ 100 $$ prizes, and fifteen $$ \$ 25 $$ prizes. Find your expected gain or loss. (Round to 2 decimal places if needed.) Expected gain or loss $$ \approx \$ $$ Enter your answer Note: Represent a gain with a positive answer and a loss with a negative answer.

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UpStudy AI · Accepted Answer

Let the net outcome be the prize amount minus the cost of the ticket. The cost of the ticket is $$\$15$$.

There are:
- 1 prize of $$\$500$$ with net gain $$\$500 - \$15 = \$485$$,
- 2 prizes of $$\$100$$ with net gain $$\$100 - \$15 = \$85$$,
- 15 prizes of $$\$25$$ with net gain $$\$25 - \$15 = \$10$$,
- The remaining $$300 - 1 - 2 - 15 = 282$$ tickets have no prize and give a net loss of $$\$15$$ (i.e. $$-\$15$$).

The probability of each outcome is:
$$
P(500) = \frac{1}{300}, \quad P(100) = \frac{2}{300}, \quad P(25) = \frac{15}{300}, \quad P(\text{no prize}) = \frac{282}{300}.
$$

The expected gain (or loss) $$E$$ is given by:
$$
E = \frac{1}{300}(485) + \frac{2}{300}(85) + \frac{15}{300}(10) + \frac{282}{300}(-15).
$$

Calculating each term:
$$
\frac{1}{300}(485) = \frac{485}{300} \approx 1.62,
$$
$$
\frac{2}{300}(85) = \frac{170}{300} \approx 0.57,
$$
$$
\frac{15}{300}(10) = \frac{150}{300} = 0.50,
$$
$$
\frac{282}{300}(-15) = -\frac{4230}{300} \approx -14.10.
$$

Now, sum these contributions:
$$
E \approx 1.62 + 0.57 + 0.50 - 14.10 \approx -11.41.
$$

Rounded to two decimal places, your expected gain or loss is approximately $$-\$11.41$$.
Your expected loss is approximately \$11.41.

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