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.
$ square ) dollars
(b) what can rita expect in the long run, after playing the game many times?
( \bigcirc ) rita can expect to gain money.
she can expect to win ( square ) dollars per toss.
( \bigcirc ) rita can expect to lose money.
she can expect to lose ( square ) dollars per toss.
( \bigcirc ) 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.

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

Step3: Simplify the result

Reduce the fraction to simplest form.
$\frac{12}{8} = 1.5$

Answer:

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