Leila 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: Leila tosses the dart once. She wins $1 if the dart lands in slice 1, $2 if the dart lands in slice 2, $5 if the dart lands in slice 3, and $8 if the dart lands in slice 4. She loses $2.50 if the dart lands in slices 5, 6, 7, or 8.

expectation = 1(/8) + 2(1/8) + 5(1/8) + 8(1/8) - 2/5(4/8)

= ....

I will gladly play this game all day long, unless there is a charge of
more than $1.80 to play the game

Ah, Leila's playing the "Let's see where this dart goes and hope for the best" game. Quite the gamble! Let's break down Leila's odds and see if we can find some humor in all these numbers.

Leila has a 1 in 8 chance of winning $1 with that number 1 slice. That's like trying to find a needle in a haystack, but hey, at least it's a golden needle!

The odds get a little better with slice 2, where Leila has a 1 in 8 chance of winning $2. That's like finding not only a needle in a haystack but also a $2 bill hidden in your old jeans pocket. Score!

Then we jump to slice 3, where Leila has a 1 in 8 chance of winning $5. Now that's more like finding a unicorn in your backyard! Majestic!

Lastly, we have slice 4, where Leila has a 1 in 8 chance of winning $8. That's like stumbling upon a pot of gold at the end of a very colorful rainbow!

But watch out for slices 5, 6, 7, and 8! Leila's got a 4 in 8 chance of losing $2.50. That's like thinking you're getting ice cream, but instead, it's just melted popsicles. Bummer!

So, Leila, you might not want to quit your day job and become a professional dart player just yet. But hey, who knows? Maybe luck will be on your side and you'll hit that coveted number 1 slice. Until then, may your throws be accurate and your sense of humor never falter!

To analyze the expected value of this game for Leila, we need to calculate the probability of the dart landing on each slice, and then multiply that by the amount she would win or lose.

Step 1: Calculate the probability of the dart landing on each slice.
- There are 8 equal-sized slices on the dartboard.
- Since the dart lands on a numbered slice at random, the probability of landing on any particular slice is 1/8 or 0.125.

Step 2: Calculate Leila's potential winnings/losses for each slice.
- If the dart lands on slice 1, Leila wins $1.
- If the dart lands on slice 2, Leila wins $2.
- If the dart lands on slice 3, Leila wins $5.
- If the dart lands on slice 4, Leila wins $8.
- If the dart lands on slice 5, 6, 7, or 8, Leila loses $2.50.

Step 3: Calculate the expected value.
- The expected value is the sum of the products of the probabilities and the corresponding winnings/losses.
- Expected Value = (Probability of slice 1) * (Amount won/lost for slice 1) + (Probability of slice 2) * (Amount won/lost for slice 2) + (Probability of slice 3) * (Amount won/lost for slice 3) + (Probability of slice 4) * (Amount won/lost for slice 4) + (Probability of slice 5) * (Amount won/lost for slice 5) + (Probability of slice 6) * (Amount won/lost for slice 6) + (Probability of slice 7) * (Amount won/lost for slice 7) + (Probability of slice 8) * (Amount won/lost for slice 8).

Let's calculate the expected value:

Expected Value = (0.125 * $1) + (0.125 * $2) + (0.125 * $5) + (0.125 * $8) + (0.125 * -$2.50) + (0.125 * -$2.50) + (0.125 * -$2.50) + (0.125 * -$2.50)
Expected Value = $0.125 + $0.25 + $0.625 + $1 + (-$0.3125) + (-$0.3125) + (-$0.3125) + (-$0.3125)
Expected Value = $1.0625 - $1 - $1.25
Expected Value = -$0.1875

Based on this analysis, the expected value for Leila's game is -$0.1875. This means that, on average, Leila is expected to lose $0.1875 per game she plays.

To determine Leila's expected value, we need to calculate the probability of each possible outcome and then multiply it by the corresponding payout or loss.

1. Probability of landing in slice 1: There is only one slice with the number 1, so the probability is 1/8.

2. Probability of landing in slice 2: Again, there is only one slice with the number 2, so the probability is 1/8.

3. Probability of landing in slice 3: Similarly, there is only one slice with the number 3, so the probability is 1/8.

4. Probability of landing in slice 4: Once again, there is only one slice with the number 4, so the probability is 1/8.

5. Probability of landing in slices 5, 6, 7, or 8: Since all these slices have an equal chance of being selected, the probability is 4/8 = 1/2.

Now, let's calculate the expected value:
(Expected Value) = (Probability of each outcome) * (Payout or loss for that outcome)

(Expected Value) = (1/8) * ($1) + (1/8) * ($2) + (1/8) * ($5) + (1/8) * ($8) + (1/2) * (-$2.50)

Simplifying the equation:
(Expected Value) = $1/8 + $2/8 + $5/8 + $8/8 - $2.50/2
(Expected Value) = $16/8 - $2.50/2
(Expected Value) = $2 - $1.25
(Expected Value) = $0.75

The expected value of Leila's game is $0.75.