Question 1191307
Here's how to calculate the expected value and variance of the net profit:

**1. Define the Random Variable:**

Let X be the random variable representing the net profit.

**2. Create the Probability Distribution:**

| Outcome | Net Profit (X) | Probability (P(X)) |
|---|---|---|
| Successful | Rs. 3,000,000 | 0.15 |
| Moderately Successful | Rs. 1,000,000 | 0.25 |
| Unsuccessful | -Rs. 1,000,000 | 0.60 |

**3. Calculate the Expected Value (Mean):**

The expected value E(X) is calculated as the sum of each possible value of X multiplied by its probability:

E(X) = (3,000,000 * 0.15) + (1,000,000 * 0.25) + (-1,000,000 * 0.60)
E(X) = 450,000 + 250,000 - 600,000
E(X) = 100,000

Therefore, the expected net profit is Rs. 100,000.

**4. Calculate the Variance:**

The variance Var(X) is calculated as the expected value of the squared difference between each value of X and the mean:

Var(X) = Σ[(X - E(X))² * P(X)]

Var(X) = (3,000,000 - 100,000)² * 0.15 + (1,000,000 - 100,000)² * 0.25 + (-1,000,000 - 100,000)² * 0.60
Var(X) = (2,900,000)² * 0.15 + (900,000)² * 0.25 + (-1,100,000)² * 0.60
Var(X) = 8,410,000,000,000 * 0.15 + 810,000,000,000 * 0.25 + 1,210,000,000,000 * 0.60
Var(X) = 1,261,500,000,000 + 202,500,000,000 + 726,000,000,000
Var(X) = 2,190,000,000,000

Therefore, the variance of the net profit is Rs. 2,190,000,000,000.

**5. Calculate the Standard Deviation:**

The standard deviation SD(X) is the square root of the variance:

SD(X) = √Var(X)
SD(X) = √2,190,000,000,000
SD(X) ≈ Rs. 1,479,864.86

**Summary of Results:**

* **Expected Value (Mean):** Rs. 100,000
* **Variance:** Rs. 2,190,000,000,000
* **Standard Deviation:** Rs. 1,479,864.86