3 A typist has probability 001 of typing a word incorrectly

3. A typist has probability 0.01 of typing a word incorrectly. (a) Use the Binomial distribution to find the probability of typing 2 or fewer words incorrectly on a page of 250 words. (b) Redo (a) using the Poisson distribution.

Solution

(a) Given X follows Binomial distribution with n=250 and p=0.01

P(X=x)=250Cx*(0.01^x)*(0.99^(250-x)) for x=0,1,2,...,250

So the probability is

P(X<=2) = P(X=0)+P(X=1)+P(X=2)

=250C0*(0.01^0)*(0.99^(250-0))+...+250C2*(0.01^2)*(0.99^(250-2))

=0.543169

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(b) Given X follows Poisson distribution with mean=n*p=250*0.01 =2.5

P(X=x)=(2.5^x)*exp(-2.5)/x! for x=0,1,2,...

So the probability is

P(X<=2) = P(X=0)+P(X=1)+P(X=2)

=(2.5^0)*exp(-2.5)/1+(2.5^1)*exp(-2.5)/1+(2.5^2)*exp(-2.5)/2

=0.5438131