Answer to Question #5219 in Statistics and Probability for cash

Question

Three stores have 8, 12, and 15 employees of whom 3, 8, and 7, respectively, are women.

(i) A store is chosen at random and from that store an employee is chosen at random. If this employee is a woman, what is the probability she came from the store with 12 employees?

(ii) If a second employee is chosen from the same store in (i), what is the probability that a woman will be chosen? Assume that the employee chosen in (i) was a woman and we don’t know which store she came from.

[7 marks]

Answer 1

Consider the events:
A - the store with 8 employees is chosen,
B - the
store with 12 employees is chosen,
A - the store with 15 employees is
chosen,
W - employee is a woman.

i)
We need to find the conditional
probability

P(B|W) = |using Bayes' theorem| = (P(W|B)*P(B)) / (P(W|A) +
P(W|B) + P(W|C));
P(B|W) = (8/12*1/3) / (3/8 + 8/12 + 7/15) =
80/543.

ii)
Let's first calculate
P(A|W) and P(C|W) like in part
i):
P(A|W) = (3/8*1/3) / (3/8 + 8/12 + 7/15) = 45/543.
P(C|W) = (7/15*1/3)
/ (3/8 + 8/12 + 7/15) = 56/543.

Using the law of total probability, we
get:
P(W) = P(W|A)*P{the first woman was from A} + P(W|B)*P{the first woman
was from B} +
P(W|C)*P{the first woman was from C} = 2/8*45/543 +
7/12*80/543 + 6/15*56/543 = 1/543*(45/4 + 140/3 + 112/5) = 4819/(543*60) =
0.148...