Find the probability that two such proofreaders working independently will miss at least one error in a work that contains four errors.

Question

A professional proofreader has a 98% chance of detecting an error in a piece of written work (other than misspellings, double words, and similar errors that are machine detected). A work contains four errors. Find the probability that two such proofreaders working independently will miss at least one error in a work that contains four errors.

a) 0.0776
b) 0.9996
c) 0.0016
d) 0.9696

I tried P(all errors found)^2 and P(all errors)*2 but it is not in the given answers

Could someone give me a hint?

Answer 1

Let $p=2\%$ be the probability that one proofreader misses a particular error. The probability that this error is missed by both proofreaders then is $p^2$, and the probability that this error is detected then comes to $1-p^2$.

This allows to compute the probability that each of the four errors is detected, and finally the desired probability that at least one error remains undetected.