The incidence of an occupation disease to the workers of a factory is found to be `1//5000`. If there are 10000 workers in a factory then the probability that none of them will get the disease is

Options

A. e
B. `e^(-2)`
C. `e^(-3)`
D. `e^(4)`

Answer

To solve the problem step by step, we need to find the probability that none of the 10,000 workers in a factory will get an occupational disease, given that the incidence rate is \( \frac{1}{5000} \).

### Step 1: Identify the given values
- The incidence rate of the disease \( p = \frac{1}{5000} \)
- The total number of workers \( n = 10000 \)

### Step 2: Calculate the value of \( \lambda \)
The parameter \( \lambda \) for a Poisson distribution can be calculated using the formula:
\[ \lambda = n \times p \]
Substituting the known values:
\[ \lambda = 10000 \times \frac{1}{5000} = 2 \]

### Step 3: Use the Poisson probability formula
The probability of observing \( x \) occurrences (in this case, the number of workers getting the disease) in a Poisson distribution is given by:
\[ P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \]
We need to find the probability that none of the workers (i.e., \( x = 0 \)) will get the disease.

### Step 4: Substitute \( x = 0 \) into the formula
Substituting \( \lambda = 2 \) and \( x = 0 \):
\[ P(X = 0) = \frac{e^{-2} \cdot 2^0}{0!} \]

### Step 5: Simplify the expression
We know that:
- \( 2^0 = 1 \)
- \( 0! = 1 \)

Thus, the equation simplifies to:
\[ P(X = 0) = \frac{e^{-2} \cdot 1}{1} = e^{-2} \]

### Step 6: Conclusion
The probability that none of the 10,000 workers will get the disease is:
\[ P(X = 0) = e^{-2} \]

### Final Answer
The required probability is \( e^{-2} \).

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