College Algebra, Chapter 10, 10.4, Section 10.4, Problem 20

A psychologists needs 12 left-handed subjects for an experiment, adn she interviews 15 potential subjects. About $10 \%$ of the population is left-handed.

a.) What is the probability that exactly 12 of the potential subjects are left-handed?

b.) What is the probability that 12 or more are left-handed?

Recall that the formula for the binomial probability is given by

$C(n,r) p^r q^{n-r}$

In this case, the probability of success is $p=0.10$ and the probability of failure $q=1-p = 0.90$.

a.) The probability that exactly 12 out of 15 are left-handed is

$= C(15,12)(0.10)^{12} (0.90)^{15-12}$

$= C(15,12) (0.10)^{12} (0.90)^3$

$= 3.3117 \times 10^{-10}$

b.) The probability that 12 or more left-handed are selected is equal to

$= C(15,12)(0.10)^{12} (0.90)^{15-12} + C(15,13)(0.10)^{13} (0.90)^{15-13} + C(15,14) (0.10)^{14} (0.90)^{15-14} + C(15,15) (0.10)^{15} (0.90)^{15-15}$

$= 3.317 \times 10^{-10} + 8.505 \times 10^{-12} + 1.35 \times 10^{-13} + 1 \times 10^{-15}$

$= 3.403 \times 10^{-10}$