Question

A typist makes an error while typing a letter 0.9% of the time. There are two types of errors. Type A and B
errors occur 30% and 70% of the time, respectively, whenever an error occurs.
(a) What is the probability of no error in 15 letters?
(b) What is the probability of no type A error in 15 letters?
c) given that exactly one error has occurred in 15 letters, what is the probability that it is a type A error?

Answer 1

Let

p = probability that the typist makes an error

p = 0.9% = 0.009

A = event that Type A error has occured

P(A) = 0.3

B = event that Type B error has occured

P(B) = 0.7

a)

Let X be the number of errors in 15 letters

Then X follows Binomial (n = 15, p = 0.009)

To find P(no errors in 15 letters)

= P(X = 0)

Using Excel function BINOM.DIST

P(X = 0) = BINOM.DIST(0, 15, 0.009, FALSE)

              = 0.8732

P(no errors in 15 letters) =

b)

Let

p1 = Probability that typist makes error and it is type A error

p1 = p * P(A)

     = 0.009 * 0.3

     = 0.0027

Let Xa be the number of Type A errors in 15 letters

Then Xa follows Binomial (n = 15, p = 0.0027)

To find P(no Type A errors in 15 letters)

= P(Xa = 0)

Using Excel function BINOM.DIST

P(Xa = 0) = BINOM.DIST(0, 15, 0.0027, FALSE)

              = 0.960257

P(no type A errors in 15 letters) =

c)

To find P(it is a type A error given only 1 error has occured)

that is to find

P(Xa = 1 | X = 1)

Using Baye's conditional probability rule

Using Excel function BINOM.DIST

P(it is a type A error given only 1 error has occured) =

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