There is a room with 100 people in it. 48 are men, 20 wear glasses, and 65 are b

Question

There is a room with 100 people in it. 48 are men, 20 wear glasses, and 65 are brunettes (hair color). 5 are brunette men who wear glasses, 13 brunettes wear glasses, 8 men wear glasses, and 22 men are brunettes. If you randomly select a person form the room... If the selected person is a brunette that wears glasses, what is the probability that they are also a man? (a) (b) If a man wears glasses what are the changes they are NOT a brunette? (c) What is the probability that a man is neither a brunettes NOR wears glasses?

Explanation / Answer

Let M denote Men
Wg denote wear glasses
B denote Brunettes
Given P(Men) = 48/100 = 0.48, P(Wg) = 20/100 = 0.20 and P(B) = 65/100 = 0.65
P(M and B and Wg) = 5/100 = 0.05
P(B and Wg) = 13/100 = 0.13
P(M and Wg) = 8/100 = 0.08
P(M and B) = 22/100 = 0.22

a) P(M / B and Wg) = P(M and B and Wg) / P(B and Wg)
= 0.05 / 0.13 = 0.384615

b) P(Not B / M and Wg) = P(B' and M and Wg) / P(M and Wg)
= [ P(M and Wg) - P(B and M and Wg) ] / P(M and Wg)
= [ 0.08 - 0.05 ] / 0.08 = 0.375

c) P[ (M and B) or (M and Wg) ] = P(M and B) + P(M and Wg) - P(M and B and Wg)
= 0.22 + 0.08 - 0.05 = 0.25

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