Step 1

The binomial probability distribution is,

\(P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right)(p)^{x}(1-p)^{n-x}\)

In the formula, n denotes the number of trails, p denotes probability of success, and x denotes the number of success.

Let x be the number of women hired which follows binomial with sample size 19 and probability of success \(\displaystyle{0.5}{\left(={\frac{{{1}}}{{{2}}}}\right)}\)

Step 2

The probability of getting two or fewer women when 19 people are hired is,

\(\displaystyle{P}{\left({X}\leq{2}\right)}={P}{\left({X}={0}\right)}+{P}{\left({X}={1}\right)}+{P}{\left({X}={2}\right)}\)

\(=\left(\begin{array}{c}19\\ 0\end{array}\right)(0.5)^{0}(1-0.5)^{19-0}+\left(\begin{array}{c}19\\ 1\end{array}\right)(0.5)^{1}(1-0.5)^{19-1}+\left(\begin{array}{c}19\\ 2\end{array}\right)(0.5)^{2}(1-0.5)^{19-2}\)

\(\displaystyle={0.00000191}+{0.00003624}+{0.00032616}\)

\(\displaystyle={0.00036431}\)

Thus, the probability of getting two or fewer women when 19 people are hired is 0.00036431.