A game is played flipping three coins. If all three coins land heads up you win $15. If two coins land heads up you win $5. If one coin lands heads up you win $2. If no coins land heads up you win nothing. If you pay $4 to play, what are your expected winnings?

HHH: expectation = 15(1/8) = 1.875

2H,1T: expectation = 5(3/8) = 1.875
1H,2T expectation = 2(3/8) = 0.75
TTT expectation = 0(1/8) = 0

total expected winnings = 4.5
Since you are paying $4 to win $4.50, you would never ever see this game in a place like Vegas

To determine the expected winnings, we need to calculate the probability of each outcome and multiply it by the corresponding amount won.

Let's calculate each individual probability:

1. Probability of all three coins landing heads up:
Probability of one coin landing heads up = (1/2) * (1/2) * (1/2) = 1/8

2. Probability of two coins landing heads up:
Probability of two coins landing heads up = (1/2) * (1/2) * (1/2) = 1/8

3. Probability of one coin landing heads up:
Probability of one coin landing heads up = (1/2) * (1/2) * (1/2) = 1/8

4. Probability of no coins landing heads up:
Probability of no coin landing heads up = (1/2) * (1/2) * (1/2) = 1/8

Now let's calculate the expected winnings:

- Amount won for all three coins landing heads up = $15
Expected winnings = Probability of all three coins landing heads up * Amount won
Expected winnings = (1/8) * $15 = $1.875

- Amount won for two coins landing heads up = $5
Expected winnings = Probability of two coins landing heads up * Amount won
Expected winnings = (1/8) * $5 = $0.625

- Amount won for one coin landing heads up = $2
Expected winnings = Probability of one coin landing heads up * Amount won
Expected winnings = (1/8) * $2 = $0.25

- Amount won for no coins landing heads up = $0
Expected winnings = Probability of no coins landing heads up * Amount won
Expected winnings = (1/8) * $0 = $0

Finally, to calculate the overall expected winnings, we need to subtract the cost of playing:
Expected winnings = (Expected winnings - Cost of playing)
Expected winnings = ($1.875 + $0.625 + $0.25 + $0) - $4
Expected winnings = $2.75 - $4 = -$1.25

Therefore, the expected winnings for this game are -$1.25, which means on average, you would lose $1.25 per game.

To calculate the expected winnings, we need to find the probabilities of each outcome and multiply them by the corresponding winnings. Let's go step by step:

1. Probability of getting all three coins heads up:
Since there are two possible outcomes for each coin (heads or tails), the probability of getting all three coins heads up is calculated as:
Probability of getting heads in one coin = 1/2
Therefore, the probability of getting all three coins heads up is (1/2)^3 = 1/8.

The winnings for this outcome are $15.

2. Probability of getting exactly two coins heads up:
To get exactly two coins heads up, we can have three different combinations: HHT, HTH, or THH. Each combination has a probability of (1/2)^3 = 1/8.

The winnings for this outcome are $5.

3. Probability of getting exactly one coin heads up:
To get exactly one coin heads up, we can have three different combinations: HHT, HTH, or THH. Each combination has a probability of (1/2)^3 = 1/8.

The winnings for this outcome are $2.

4. Probability of getting no coin heads up:
Since there is only one possible outcome where no coin lands heads up, the probability is (1/2)^3 = 1/8.

The winnings for this outcome are $0 since there is no win.

Now, let's calculate the expected winnings:

Expected winnings = (Probability of outcome 1 * Winnings for outcome 1) +
(Probability of outcome 2 * Winnings for outcome 2) +
(Probability of outcome 3 * Winnings for outcome 3) +
(Probability of outcome 4 * Winnings for outcome 4)

Expected winnings = (1/8 * $15) + (3/8 * $5) + (3/8 * $2) + (1/8 * $0)

Simplifying this expression, we get:

Expected winnings = $1.875 + $1.875 + $0.75 + $0

Therefore, the expected winnings from playing the game, after deducting the $4 cost to play, is:

Expected winnings = $1.875 + $1.875 + $0.75 - $4 = $1.5

So, your expected winnings from playing this game is $1.5.