Round probabilities to four decimal places.
Suppose 60% of students enrolled in community colleges in California identify as female. A text book publishing company is going to randomly select 20 California community college students and provide them with free textbooks.
Of the 20 students randomly selected, find the probability that the number who identify as female is...
NOTE: The number of female students will be a random variable with a binomial distribution. n = 20, p = .60
a)exactly twelve.
b) thirteen or fewer.
c) fewer than sixteen.
d) between 11 and 14, inclusive.
e) 13 or more.
f) more than ten.
g) What is the expected number females receiving the free textbooks? Answer does not need to be an integer, don't round.
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To find the probabilities, we will use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where n is the number of trials, p is the probability of success, k is the number of successes, and (n choose k) is the number of ways to choose k successes from n trials.
a) For exactly twelve female students:
P(X = 12) = (20 choose 12) * (0.60)^12 * (1-0.60)^(20-12)
= 0.2527
b) For thirteen or fewer female students:
P(X ≤ 13) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 13)
We can calculate each probability separately and then sum them up:
P(X ≤ 13) = 0.0009 + 0.0065 + 0.0261 + 0.0747 + 0.1511 + 0.2335 + 0.2668 + 0.2335 + 0.1511 + 0.0747 + 0.0261 + 0.0065 + 0.0009
= 0.9522
c) For fewer than sixteen female students:
P(X < 16) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 15)
We can calculate each probability separately and then sum them up:
P(X < 16) = 0.0009 + 0.0065 + 0.0261 + 0.0747 + 0.1511 + 0.2335 + 0.2668 + 0.2335 + 0.1511 + 0.0747 + 0.0261 + 0.0065 + 0.0009
= 0.9522
d) For between 11 and 14, inclusive, female students:
P(11 ≤ X ≤ 14) = P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14)
We can calculate each probability separately and then sum them up:
P(11 ≤ X ≤ 14) = 0.1535 + 0.2527 + 0.2508 + 0.1790
= 0.836
e) For 13 or more female students:
P(X ≥ 13) = 1 - P(X < 13) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 12))
We can calculate each probability separately, sum them up, and subtract from 1:
P(X ≥ 13) = 1 - (0.0009 + 0.0065 + 0.0261 + 0.0747 + 0.1511 + 0.2335 + 0.2668 + 0.2335 + 0.1511 + 0.0747 + 0.0261 + 0.0065)
= 1 - 0.9965
= 0.0035
f) For more than ten female students:
P(X > 10) = 1 - P(X ≤ 10) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 10))
We can calculate each probability separately, sum them up, and subtract from 1:
P(X > 10) = 1 - (0.0009 + 0.0065 + 0.0261 + 0.0747 + 0.1511 + 0.2335 + 0.2668 + 0.2335 + 0.1511 + 0.0747 + 0.0261)
= 1 - 0.9930
= 0.0070
g) To find the expected number of female students (E(X)), we can use the formula:
E(X) = n * p
= 20 * 0.60
= 12
So, the expected number of female students receiving the free textbooks is 12.
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