Answer:
(a) The probability of exactly 1 page has error is 0.271.
(b) The probability that there are at most 3 pages has error is 0.857.
Step-by-step explanation:
Let X = number of typos.
The probability of a typo is, P (X) = p = 0.005.
The number of pages in the novel is, n = 400.
The random variable X follows a Binomial distribution with parameter n and p.
But as the probability is very small and the sample size is too large we can use Poisson distribution to approximate the binomial distribution.
This distribution has parameter, [tex]\lambda=np=400\times0.005=2[/tex].
The probability mass function of the Poisson distribution is:
[tex]P(X=x)=\frac{e^{-2}2^{x}}{x!} ;\ x=0,1,2,...[/tex]
(a)
Compute the probability of exactly 1 page has error as follows:
[tex]P(X=1)=\frac{e^{-2}2^{1}}{1!} =\frac{0.13534\times2}{1} =0.27068\approx0.271[/tex]
Thus, the probability of exactly 1 page has error is 0.271.
(b)
Compute the probability that there are at most 3 pages has error as follows:
P (X ≤ 3) = P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)
[tex]=\frac{e^{-2}2^{0}}{0!}+\frac{e^{-2}2^{1}}{1!}+\frac{e^{-2}2^{2}}{2!}+\frac{e^{-2}2^{3}}{3!}\\=0.13534+0.27067+0.27067+0.18045\\=0.85713\\\approx0.857[/tex]
Thus, the probability that there are at most 3 pages has error is 0.857.
