Answer :
Final answer:
To find the probability that exactly eleven people have type A blood out of sixteen randomly chosen people, use the binomial probability formula. The probability that more than eleven people have type A blood can be found by summing up the probabilities of exactly twelve to sixteen people having type A blood.
Explanation:
To find the probability that exactly eleven people have type A blood, we will use the binomial probability formula. There are 40 people with type A blood and 16 people chosen, and we want to find the probability of choosing exactly eleven people with type A blood. The probability of choosing a person with type A blood is 40/100, and the probability of choosing a person without type A blood is 60/100. Using the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where n is the number of trials (16), k is the number of successful trials (11), p is the probability of success (40/100), and C(n, k) is the binomial coefficient. Plugging in the values:
P(X = 11) = C(16, 11) * (40/100)^11 * (60/100)^(16-11)
Simplifying the expression gives:
P(X = 11) ≈ 0.069
To find the probability that more than eleven people have type A blood, we can find the probabilities of exactly twelve people, thirteen people, and so on, up to sixteen people having type A blood, and then sum up these probabilities:
P(X > 11) = P(X = 12) + P(X = 13) + ... + P(X = 16)
Using the same formula as above:
P(X > 11) = Σ P(X = k) for k = 12 to 16
Calculating the individual probabilities and summing them up, we find:
P(X > 11) ≈ 0.088
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