you flip a coin 3 times. each time you get head you get 5 coins, tale you pay 2 coins, and every flip you pay a coin. caluculate the expected value and the variance of the revenue

To solve this problem, we need to calculate the expected value (mean) and variance of the revenue obtained from flipping a coin 3 times under the given conditions:

- **Head (H):** You gain 5 coins.
- **Tail (T):** You lose 2 coins.
- **Each flip costs 1 coin** (so, total cost for 3 flips is 3 coins).

### Step-by-step Solution:

#### Step 1: Define the Random Variables

Let's define the random variable \( X \) as the net revenue from flipping the coin once (not counting the 1-coin cost for each flip initially).

1. **Possible outcomes for one flip:**
   - **Head (H):** Gain 5 coins.
   - **Tail (T):** Lose 2 coins.

2. **Net revenue for one flip after deducting the cost:**
   - **Head (H):** \(5 - 1 = 4\) coins.
   - **Tail (T):** \(-2 - 1 = -3\) coins.

#### Step 2: Calculate the Expected Value \( E(X) \)

The probability of getting a Head or Tail in each flip is \( \frac{1}{2} \).

\[
E(X) = P(\text{Head}) \cdot \text{Revenue from Head} + P(\text{Tail}) \cdot \text{Revenue from Tail}
\]

\[
E(X) = \frac{1}{2} \cdot 4 + \frac{1}{2} \cdot (-3)
\]

\[
E(X) = 2 - 1.5 = 0.5
\]

#### Step 3: Calculate the Expected Value of the Total Revenue

Since you flip the coin 3 times, let \( Y \) represent the total revenue from all 3 flips:

\[
E(Y) = 3 \cdot E(X) = 3 \cdot 0.5 = 1.5
\]

#### Step 4: Calculate the Variance \( V(X) \)

Variance measures the spread of the possible outcomes. First, we need to compute \( E(X^2) \):

\[
E(X^2) = \frac{1}{2} \cdot (4^2) + \frac{1}{2} \cdot (-3^2)
\]

\[
E(X^2) = \frac{1}{2} \cdot 16 + \frac{1}{2} \cdot 9
\]

\[
E(X^2) = 8 + 4.5 = 12.5
\]

Now, the variance \( V(X) \) is given by:

\[
V(X) = E(X^2) - (E(X))^2
\]

\[
V(X) = 12.5 - (0.5)^2
\]

\[
V(X) = 12.5 - 0.25 = 12.25
\]

#### Step 5: Calculate the Variance of the Total Revenue

For 3 independent flips, the variance of the total revenue \( Y \) is:

\[
V(Y) = 3 \cdot V(X) = 3 \cdot 12.25 = 36.75
\]

### Conclusion:

- **Expected value of the total revenue after 3 flips ( \( E(Y) \) ):** 1.5 coins.
- **Variance of the total revenue after 3 flips ( \( V(Y) \) ):** 36.75 coins\(^2\).

Would you like more details or have any questions?

### Related Questions:

1. How would the expected value and variance change if the number of flips increased to 5?
2. What is the probability of getting a net positive revenue after 3 flips?
3. How does changing the payout for tails affect the variance?
4. What happens to the expected value if the cost per flip changes?
5. How would the expected value and variance differ if the coin was biased (not fair)?

### Tip:
When calculating the variance for independent random variables, remember that the variance of their sum is the sum of their variances.

Learn how to calculate the expected value and variance of revenue from flipping a coin 3 times, where each head gains 5 coins, each tail loses 2 coins, and each flip costs 1 coin. Understand the probability theory and formulas used in this problem.

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