Step 1

Solution:

Let X be the number of teenage customers will have part-time jobs.

From the given information, probability that a teenage customer receive their spending money from part-time job is 0.40 and \(\displaystyle{n}={5}\).

Step 2

Here, teenagers are independent and probability of success is constant. Hence, X follows binomial distribution with parameters \(\displaystyle{n}={5}\ {\quad\text{and}\quad}\ {p}={0.40}\).

The probability mass function of binomial random variable X is

\[P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right) p^{x}(1-p)^{n-x}; x=0,1,.......,n\]

Step 3

1. The probability that at least 3 of them will have part-time jobs is

\(\displaystyle{P}{\left({X}\geq{3}\right)}={1}-{P}{\left({X}{<}{3}\right)}\)

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

\(\displaystyle={1}-{0.6826}\) [Using the excel function =BINOM.DIST (2,5,0.4, TRUE)]

\(\displaystyle={0.3174}\)

Thus, the probability that at least 3 of them will have part-time jobs is 0.3174.

Step 4

2. The probability that no more than 4 will have part-time jobs is

\(\displaystyle{P}{\left({X}\leq{4}\right)}={0.9898}\) [Using the excel function =BINOM.DIST (4,5,0.4, TRUE)]

Thus, the probability that no more than 4 will have part-time jobs is 0.9898.

Step 5

3. The probability that less than 5 but greater than 3 will have part-time jobs is

\(\displaystyle{P}{\left({3}{<}{X}{<}{5}\right)}={P}{\left({X}={4}\right)}\)

\[=\left(\begin{array}{c}5\\ 4\end{array}\right) 0.40^{4} (1-0.40)^{5-4}\]

\(\displaystyle={0.0768}\)

Thus, the probability that less than 5 but greater than 3 will have part-time jobs is 0.0768.