Question 1206867: In a city school, 60% of students have blue eyes, 55% have dark hair, and 20% have neither blue eyes nor dark hair. Determine the probability that a randomly selected student will have blue eyes and dark hair.
Answer by ikleyn(52770)   (Show Source):
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In a city school, 60% of students have blue eyes, 55% have dark hair,
and 20% have neither blue eyes nor dark hair. Determine the probability
that a randomly selected student will have blue eyes and dark hair.
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Since 20% have neither blue eyes nor dark hair, we conclude that
100 -20% = 80% are those who either have blue eyes OR dark hair (or both).
So, now we have that 60% of students have blue eyes P(B) = 0.6,
55% have dark hair P(D) = 0.55, and their union (B U D) is 80%: P(B U D) = 0.8.
Having it and using basic formula of probability, we write
P(B U D) = 0.8,
P(B) + P(D) - P(B and D) = 0.8,
0.6 + 0.55 - P(B and D) = 0.8,
0.6 + 0.55 - 0.8 = P(B and D),
P(B and D) = 0.35.
Therefore, the probability that a randomly selected student will have blue eyes and dark hair is
P(B and D) = 0.35 = 35%. ANSWER
Solved.
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