Number of accidents on a national highway each day is a poisson variable with an average of three accidents per a day. Find the probability that no accidents will occur on a given day.

Answer

To solve the problem of finding the probability that no accidents will occur on a given day when the number of accidents follows a Poisson distribution with an average of 3 accidents per day, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Poisson Distribution Parameters**:
- The average number of accidents per day (λ) is given as 3. This is our parameter for the Poisson distribution.

2. **Recall the Probability Mass Function of Poisson Distribution**:
- The probability mass function (PMF) for a Poisson random variable is given by:
\[ P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \]
Where:
- \( P(X = x) \) is the probability of observing \( x \) events (accidents in this case).
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
- \( \lambda \) is the average rate (3 in this case).
- \( x \) is the number of occurrences (accidents).

3. **Set the Value of x**:
- We need to find the probability of no accidents occurring in a day, which means \( x = 0 \).

4. **Substitute the Values into the PMF**:
- Substitute \( \lambda = 3 \) and \( x = 0 \) into the PMF formula:
\[ P(X = 0) = \frac{e^{-3} \cdot 3^0}{0!} \]

5. **Simplify the Expression**:
- Calculate \( 3^0 \) which is 1, and \( 0! \) which is also 1:
\[ P(X = 0) = \frac{e^{-3} \cdot 1}{1} = e^{-3} \]

6. **Calculate the Value of \( e^{-3} \)**:
- Using a calculator, we find:
\[ e^{-3} \approx 0.04979 \]

7. **Final Answer**:
- Therefore, the probability that no accidents will occur on a given day is approximately:
\[ P(X = 0) \approx 0.04979 \]

### Summary:
The probability that no accidents will occur on a given day is approximately **0.04979**.