Question

A large disaster cleaning company estimates that 30 percent of the jobs it bids on are finished within the bid time. Looking at a random sample of 8 jobs that it has contracted, calculate the probability that exactly 4 of the jobs were not completed within the bid time.

Answer 1

Answer:

There is a 13.61% probability that exactly 4 of the jobs were not completed within the bid time.

Step-by-step explanation:

For each job, there are only two possible outcomes. Either they are completed on time, or they are not. This means that we use the binomial probability distribution to solve this problem.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

Looking at a random sample of 8 jobs that it has contracted, calculate the probability that exactly 4 of the jobs were not completed within the bid time.

There are 8 jobs, so $n = 8$.

30% of them are finished on time, which means that 70% are not completed within the bid time. This means that $p = 0.7$

The problems asks for P(X = 4). So

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 4) = C_{8,4}.(0.7)^{4}.(0.3)^{4} = 0.1361$

There is a 13.61% probability that exactly 4 of the jobs were not completed within the bid time.