Question

rita is playing a game of chance in which she tosses a dart into a rotating dartboard with 8 equal - sized slices numbered 1 through 8. the dart lands on a numbered slice at random. this game is this: rita tosses the dart once. she wins $1 if the dart lands in slice 1, $3 if the dart lands in slice 2, $5 if the dart lands in slice 3, $8 if the dart lands in slice 4, and $10 if the dart lands in slice 5. she loses $5 if the dart lands in slices 6, 7, or 8. (if necessary, consult a list of formulas.) (a) find the expected value of playing the game. dollars (b) what can rita expect in the long run, after playing the game many times? ○ rita can expect to gain money. she can expect to win dollars per toss. ○ rita can expect to lose money. she can expect to lose dollars per toss. ○ rita can expect to break even (neither gain nor lose money).

Explanation:

Step1: Define probabilities & payouts

Each slice has probability $\frac{1}{8}$.
Payouts:
Slice1: $\$1$, Slice2: $\$3$, Slice3: $\$5$, Slice4: $\$8$, Slice5: $\$10$, Slices6-8: $-\$5$

Step2: Calculate expected value

Multiply each payout by its probability, sum all terms.

$$\begin{align*} E(X) &= \frac{1}{8}(1) + \frac{1}{8}(3) + \frac{1}{8}(5) + \frac{1}{8}(8) + \frac{1}{8}(10) + 3\times\frac{1}{8}(-5) \\ &= \frac{1+3+5+8+10-15}{8} \\ &= \frac{12}{8} \end{align*}$$

Step3: Simplify the result

Reduce the fraction to lowest terms.
$\frac{12}{8} = \frac{3}{2} = 1.5$

Answer:

(a) $\boldsymbol{1.5}$
(b) Rita can expect to gain money.
She can expect to win $\boldsymbol{1.5}$ dollars per toss.