Question

a warehouse employs 24 workers on first shift, 19 workers on second shift, and 10 workers on third shift. eight workers are chosen at random to be interviewed about the work environment. find the probability of choosing exactly two second shift workers and two third shift workers.
the probability of choosing two second shift workers and two third shift workers is 0.332 (round to three decimal places as needed.)

Explanation:

Step1: Calculate total number of workers

The total number of workers is $24 + 19+10=53$.

Step2: Calculate number of ways to choose 8 workers in total

The number of ways to choose 8 workers out of 53 is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 53$ and $r = 8$. So, $C(53,8)=\frac{53!}{8!(53 - 8)!}=\frac{53!}{8!45!}=\frac{53\times52\times51\times50\times49\times48\times47\times46}{8\times7\times6\times5\times4\times3\times2\times1}$.

Step3: Calculate number of ways to choose 2 second - shift workers

There are 19 second - shift workers, and the number of ways to choose 2 of them is $C(19,2)=\frac{19!}{2!(19 - 2)!}=\frac{19\times18}{2\times1}= 171$.

Step4: Calculate number of ways to choose 2 third - shift workers

There are 10 third - shift workers, and the number of ways to choose 2 of them is $C(10,2)=\frac{10!}{2!(10 - 2)!}=\frac{10\times9}{2\times1}=45$.

Step5: Calculate number of ways to choose remaining workers from first - shift

We have already chosen $2 + 2=4$ workers. We need to choose $8-(2 + 2)=4$ workers from the 24 first - shift workers. The number of ways to do this is $C(24,4)=\frac{24!}{4!(24 - 4)!}=\frac{24\times23\times22\times21}{4\times3\times2\times1}=10626$.

Step6: Calculate number of favorable cases

The number of favorable cases (choosing 2 second - shift, 2 third - shift and 4 first - shift workers) is the product of the number of ways of choosing workers from each shift, i.e., $C(19,2)\times C(10,2)\times C(24,4)=171\times45\times10626 = 171\times478170=81767070$.

Step7: Calculate the probability

The probability $P$ is the number of favorable cases divided by the number of total cases. So $P=\frac{C(19,2)\times C(10,2)\times C(24,4)}{C(53,8)}=\frac{81767070}{2349068880}\approx0.348$.

Answer:

$0.348$