A person gets Rs. 2 if head comes and loses Rs. 5 when a tail comes then how much money can he expect in the long run per game, when an unbiased coin in tossed twice.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Game
A person earns Rs. 2 for each head and loses Rs. 5 for each tail when tossing an unbiased coin twice. We need to find the expected amount of money he can expect to earn or lose in the long run per game.

### Step 2: Define the Random Variable
Let \( X \) denote the number of heads that come up in the two tosses of the coin. The possible values for \( X \) are 0, 1, and 2.

### Step 3: Calculate the Probabilities
We need to find the probabilities of getting 0, 1, or 2 heads when tossing the coin twice.

1. **Probability of 0 heads (X = 0)**:
- This occurs when both tosses are tails (TT).
- Probability: \( P(X = 0) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \).

2. **Probability of 1 head (X = 1)**:
- This occurs when one toss is a head and the other is a tail (HT or TH).
- There are 2 favorable outcomes (HT, TH).
- Probability: \( P(X = 1) = 2 \times \left(\frac{1}{2} \times \frac{1}{2}\right) = 2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \).

3. **Probability of 2 heads (X = 2)**:
- This occurs when both tosses are heads (HH).
- Probability: \( P(X = 2) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \).

### Step 4: Calculate the Expected Value of X
The expected value \( E(X) \) is calculated using the formula:
\[ E(X) = \sum (x \cdot P(X = x)) \]
Substituting the values:
\[ E(X) = 0 \cdot P(X = 0) + 1 \cdot P(X = 1) + 2 \cdot P(X = 2) \]
\[ E(X) = 0 \cdot \frac{1}{4} + 1 \cdot \frac{1}{2} + 2 \cdot \frac{1}{4} \]
\[ E(X) = 0 + \frac{1}{2} + \frac{2}{4} = 0 + \frac{1}{2} + \frac{1}{2} = 1 \]

### Step 5: Calculate the Expected Money Earned
Now, we need to calculate the expected money based on the expected number of heads:
- For each head, the person earns Rs. 2.
- For each tail, the person loses Rs. 5.

In two tosses, if the expected number of heads is 1, then the expected number of tails is also 1 (since there are 2 tosses in total).

Calculating the expected money:
- Money earned from heads: \( 1 \text{ head} \times 2 \text{ Rs/head} = 2 \text{ Rs} \)
- Money lost from tails: \( 1 \text{ tail} \times 5 \text{ Rs/tail} = 5 \text{ Rs} \)

Total expected money:
\[ \text{Expected Money} = \text{Money from heads} - \text{Money lost from tails} = 2 - 5 = -3 \text{ Rs} \]

### Conclusion
The expected amount of money the person can expect to lose in the long run per game is Rs. -3.

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