Explanation:
there are
5+6+8 = 19 employees in total.
5+8 = 13 are not technicians.
a probability is always desired outcomes over totally possible outcomes.
when we pick 5, each pick has its own probabilities, because we are not returning the picked people for the following picking rounds.
so, for the first round we have all people available.
the probability to pick a non-technician is
13/19 (there are 13 mon-technicians out of 19)
for the second round we have now 18 people to pick from, and only 12 non-technicians.
the probability to pick one is then
12/18 = 2/3
for the third round we have only 17 people, 11 non-technicians.
the probability
11/17
for the fourth round : 16 people, 10 non- technicians.
10/16 = 5/8
for the 5th round : 15 people, 9 non-technicians.
9/15 = 3/5
now we combine all 5 picks into 1 event :
13/19 × 2/3 × 11/17 × 5/8 × 3/5 = 4290/38760 =
= 715/6460 = 143/1292 = 11×13 / (4×17×19)
that means it cannot be further refined.
the probability of not having a technician in the picked group of 5 is
143/1292 = 0.110681115...