In a book of 520 pages, 390 typo-graphical errors occur. Assuming Poisson law forthe number of errors per page, find tile probability that a random sample of 5 pages will containno error
Question
In a book of 520 pages, 390 typo-graphical errors occur. Assuming Poisson law forthe number of errors per page, find tile probability that a random sample of 5 pages will containno error
Solution
To find the probability that a random sample of 5 pages will contain no errors, we can use the Poisson distribution. The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time or space.
In this case, we assume that the number of typographical errors per page follows a Poisson distribution. The average number of errors per page can be calculated by dividing the total number of errors (390) by the total number of pages (520).
Average number of errors per page = 390 / 520 = 0.75
Now, we can use the Poisson distribution formula to calculate the probability of having no errors in a sample of 5 pages. The formula is:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- P(X = k) is the probability of having exactly k events
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the average number of events (in this case, the average number of errors per page)
- k is the number of events (in this case, 0)
Plugging in the values, we get:
P(X = 0) = (e^(-0.75) * 0.75^0) / 0!
Since any number raised to the power of 0 is 1, and 0! is also 1, the formula simplifies to:
P(X = 0) = e^(-0.75)
Using a calculator, we can find that e^(-0.75) is approximately 0.4724.
Therefore, the probability that a random sample of 5 pages will contain no errors is approximately 0.4724, or 47.24%.
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