In a book of 450 pages, there are 400 typographical eroors. Assuming that the number of errors per page follow the Poisson law, find the probability that a random sample of 5 pages will contain no typographical error.

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the average number of typographical errors per page (λ)
Given:
- Total number of pages = 450
- Total number of typographical errors = 400

Using the formula for λ (lambda) in a Poisson distribution:
\[ \lambda = \frac{\text{Total errors}}{\text{Total pages}} = \frac{400}{450} \]
Calculating λ:
\[ \lambda = \frac{400}{450} = \frac{8}{9} \]

### Step 2: Use the Poisson probability formula
The Poisson probability formula is given by:
\[ P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \]
Where:
- \( e \) is the base of the natural logarithm (approximately 2.71828),
- \( x \) is the number of occurrences (in this case, typographical errors),
- \( \lambda \) is the average number of occurrences.

### Step 3: Calculate the probability of no typographical errors (x = 0)
We need to find the probability that a random sample of 5 pages will contain no typographical errors. For no errors, \( x = 0 \):
\[ P(X = 0) = \frac{e^{-\lambda} \lambda^0}{0!} \]
Substituting \( \lambda = \frac{8}{9} \):
\[ P(X = 0) = \frac{e^{-\frac{8}{9}} \left(\frac{8}{9}\right)^0}{0!} \]
Since \( \left(\frac{8}{9}\right)^0 = 1 \) and \( 0! = 1 \):
\[ P(X = 0) = e^{-\frac{8}{9}} \]

### Step 4: Calculate the probability for 5 pages
Since we are looking for the probability that all 5 pages contain no typographical errors, we will raise the probability for one page to the power of 5:
\[ P(\text{No errors in 5 pages}) = \left(e^{-\frac{8}{9}}\right)^5 = e^{-\frac{40}{9}} \]

### Final Answer
Thus, the probability that a random sample of 5 pages will contain no typographical errors is:
\[ P(\text{No errors in 5 pages}) = e^{-\frac{40}{9}} \]

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