A source in the Internal Revenue Service has stated that historically 90% of federal tax returns filed are free of arithmetic errors. A random sample of 25 returns are selected and checked carefully for arithmetic errors. Assuming independence, find the probability that

Question

a. all 25 returns are free of arithmetic errors.
b. at most 23 returns are free of arithmetic errors.
c. more than 17 are free of arithmetic errors

Answer 1

This problem has the following characteristics:
1. Number of trials is fixed (25) and known.
2. The probability of success is constant (0.9) throughout the trials and known.
3. The tests are independent
4. Each test is Bernoulli (i.e. true/false, 0/1)
These criteria indicate the binomial distribution, with
n=25, p=0.9, q=(1-0.9)=0.1
with
P(X=k)=C(n,k)p^kq^n-k

a.
Calculate P(X=25)

b.
calculate p=P(X=24)+P(X=25)
Probability for at most 23
=1-(P(X=24)+P(X=25))

c.
calculate
25
∑ P(X=k)
k=18

Answer 2

.072

Answer 3

1.075