You win $4 if you flip a coin twice and get tails both times. For any other outcome you get nothing. What is the expected value?

Well, let's do some coin flipping math! There are four possible outcomes when flipping a coin twice: heads-heads, heads-tails, tails-heads, and tails-tails. Since we only win $4 if we get tails both times, and we get nothing for any other outcome, the only outcome that matters is getting tails twice.

The probability of getting tails on a single coin flip is ½ (assuming it's a fair coin and not one of those sneaky trick coins). So, the probability of getting tails on both flips is ½ times ½, which gives us ¼.

Now that we know the probability of getting tails twice, we can calculate the expected value. Since we win $4 in only one out of the four possible outcomes, the expected value is:

(¼) × ($4) + (¾) × ($0) = $1.

So, the expected value of this game is $1. But hey, don't worry, even if you don't win the $4, at least you gained some math skills!

To calculate the expected value, we need to multiply the value of each outcome by its probability and sum them up.

Given that there are four possible outcomes when flipping a coin twice: HH, HT, TH, and TT, where H represents heads and T represents tails, we can assign the following values to each outcome:

HH: $0 (No win)
HT: $0 (No win)
TH: $0 (No win)
TT: $4 (Win)

The probabilities of these outcomes can be calculated using the concept of independent events. Since each flip of a coin is independent, the probability of getting tails on any single flip is 1/2. Therefore, the probabilities for each outcome are:

HH: (1/2) * (1/2) = 1/4
HT: (1/2) * (1/2) = 1/4
TH: (1/2) * (1/2) = 1/4
TT: (1/2) * (1/2) = 1/4

Now, we can calculate the expected value:

Expected Value = (Value of HH * Probability of HH) + (Value of HT * Probability of HT) + (Value of TH * Probability of TH) + (Value of TT * Probability of TT)

Expected Value = ($0 * 1/4) + ($0 * 1/4) + ($0 * 1/4) + ($4 * 1/4)
Expected Value = $0 + $0 + $0 + $1
Expected Value = $1

Therefore, the expected value from flipping the coin twice is $1.

To calculate the expected value, we multiply the outcome of each event by its probability and then sum them up.

In this case, we have two possible outcomes: getting tails twice or any other outcome.

Let's calculate the probability of each outcome:
- Getting tails twice: To calculate the probability of independent events, we multiply the probabilities of each event. The probability of getting tails on a fair coin flip is 1/2, so the probability of getting tails twice is (1/2) * (1/2) = 1/4.

- Any other outcome: Since there are only two possible outcomes (tails twice or any other outcome), the probability of any other outcome is 1 - 1/4 = 3/4.

Now let's calculate the expected value:
- Getting tails twice: The outcome in this case is winning $4, so the contribution to the expected value is (1/4) * $4 = $1.

- Any other outcome: The outcome in this case is winning $0, so the contribution to the expected value is (3/4) * $0 = $0.

Summing up the contributions, we have $1 + $0 = $1.

Therefore, the expected value in this scenario is $1.

1/4 for winning and 3/4 for losing. You don't mention investment.