Answer:
Explanation:
(a) Here is a tree diagram to model the situation:
P(red hair)=0.025
/ \
P(brown eyed | red hair)=0.22 P(not brown eyed | red hair)=0.78
/ \
P(brown eyed | not red hair)=0.65 P(not brown eyed | not red hair)=0.35
(b) The probability that a randomly selected person is brown eyed is equal to the sum of the probabilities of being brown eyed given red hair and being brown eyed given not red hair, weighted by the probabilities of having red hair and not having red hair:
P(brown eyed) = P(brown eyed | red hair) * P(red hair) + P(brown eyed | not red hair) * P(not red hair)
= 0.22 * 0.025 + 0.65 * (1 - 0.025)
= 0.6385
Therefore, the probability that a randomly selected person is brown eyed is approximately 0.6385, or 63.85%.
(c) The probability that a randomly selected person has red hair given the person selected isn't brown eyed can be calculated using Bayes' theorem:
P(red hair | not brown eyed) = P(not brown eyed | red hair) * P(red hair) / P(not brown eyed)
To calculate the denominator, we need to use the law of total probability, which states that the probability of an event is equal to the sum of the probabilities of that event given each possible condition:
P(not brown eyed) = P(not brown eyed | red hair) * P(red hair) + P(not brown eyed | not red hair) * P(not red hair)
= 0.78 * 0.025 + 0.35 * (1 - 0.025)
= 0.3685
Substituting this and the other given probabilities into Bayes' theorem, we get:
P(red hair | not brown eyed) = 0.78 * 0.025 / 0.3685
≈ 0.0529
Therefore, the probability that a randomly selected person has red hair given the person selected isn't brown eyed is approximately 0.0529, or 5.29%.
(d) The probability that a sample of 20 people will have at least 1 person with red hair can be calculated using the complement rule, which states that the probability of an event happening is equal to 1 minus the probability of the event not happening. In this case, the event of interest is that none of the 20 people have red hair, so the probability of this event happening is:
P(none have red hair) = (1 - 0.025)^20
≈ 0.3585
Therefore, the probability that at least 1 person in the sample has red hair is:
P(at least 1 has red hair) = 1 - P(none have red hair)
≈ 0.6415
So, the probability that a sample of 20 people will have at least 1 person with red hair is approximately 0.6415, or 64.15%.