A player tosses two coins. He wins Rs. 1 if 1 head appears, Rs 2 if 2 heads appear. But he lose Rs 5 if no head appears. The mean of the prized money is

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `-(1)/(4)`
D. `(1)/(5)`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the outcomes of tossing two coins.
When two coins are tossed, the possible outcomes are:
1. HH (both heads)
2. HT (head on the first coin, tail on the second)
3. TH (tail on the first coin, head on the second)
4. TT (both tails)

### Step 2: Determine the prize money associated with each outcome.
- If both coins show heads (HH), the player wins Rs. 2.
- If one head appears (HT or TH), the player wins Rs. 1.
- If no heads appear (TT), the player loses Rs. 5.

### Step 3: Calculate the probabilities of each outcome.
- Probability of getting 2 heads (HH): There is 1 favorable outcome (HH) out of 4 total outcomes. Thus, P(2 heads) = 1/4.
- Probability of getting 1 head (HT or TH): There are 2 favorable outcomes (HT, TH) out of 4 total outcomes. Thus, P(1 head) = 2/4 = 1/2.
- Probability of getting no heads (TT): There is 1 favorable outcome (TT) out of 4 total outcomes. Thus, P(no heads) = 1/4.

### Step 4: Define the random variable X and its corresponding values.
Let X be the random variable representing the prize money:
- X = 2 (for 2 heads)
- X = 1 (for 1 head)
- X = -5 (for no heads)

### Step 5: Calculate the expected value (mean) of the random variable X.
The expected value (mean) E(X) is calculated using the formula:
\[ E(X) = \sum (X_i \cdot P(X_i)) \]
Where \( X_i \) are the values of the random variable and \( P(X_i) \) are their corresponding probabilities.

Calculating E(X):
- For 2 heads: \( 2 \cdot \frac{1}{4} = \frac{2}{4} \)
- For 1 head: \( 1 \cdot \frac{1}{2} = \frac{2}{4} \)
- For no heads: \( -5 \cdot \frac{1}{4} = -\frac{5}{4} \)

Now, summing these values:
\[ E(X) = \frac{2}{4} + \frac{2}{4} - \frac{5}{4} \]
\[ E(X) = \frac{2 + 2 - 5}{4} = \frac{-1}{4} \]

### Final Answer:
The mean of the prize money is \( -\frac{1}{4} \) Rs.

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