# A survey from Teenage Research Unlimited found that 40% of teenage con

A survey from Teenage Research Unlimited found that 40% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random:
1. Find the probability that at least 3 of them will have part-time jobs.
2. Find the probability that no more than 4 will have part-time jobs.
3. Find the probability that less than 5 but greater than 3 will have part-time jobs.

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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.