## Problem 1016

97% of Dr. William’s patients end up with 20-30 vision or better. Find the probability that exactly 3 of her next patients end up with 20-30 vision or better.

**Solution:-**

Assuming the events are independent, than the probability of obtaining exactly x successes, P(x), in n independent trails is given by P(x) = (_{n}C_{x}) p^{x}q^{n-x} where p is the probability of success on a single trial and q is the probability of failure on a single trial.

We will consider a ‘success’ a patient ending with 20-30 vision or better.

The probability the patient’s vision is a success is 0.97.

The probability the patient’s vision is a failure is thus 1 – 0.97, or 0.03.

Thus, n = 4, x = 3, p = 0.97, and q = 0.03.

Substitute the values.

P(3) = (_{4}C_{3})(0.97)^{3}(0.03)^{1}

=4(0.912673)(0.0300)

= 0.1095