At a local community college, 50% of students who enter the college as freshmen go on to graduate. Ten freshmen are randomly selected, what is the probability that none of them graduates from the local community college (round your answer to four decimal places)
Question
At a local community college, 50% of students who enter the college as freshmen go on to graduate. Ten freshmen are randomly selected, what is the probability that none of them graduates from the local community college (round your answer to four decimal places)
Solution
This is a binomial probability problem. The formula for binomial probability is:
P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
where:
- P(X=k) is the probability of k successes in n trials,
- C(n, k) is the number of combinations of n items taken k at a time,
- p is the probability of success on any given trial,
- n is the total number of trials, and
- k is the number of successes we are interested in.
In this case:
- The "success" is a student not graduating,
- The probability of success (p) is 0.5 (since 50% of students do not graduate),
- The total number of trials (n) is 10 (since we are selecting 10 students), and
- The number of successes we are interested in (k) is 0 (since we want to know the probability that none of them graduates).
So, plugging in these values, we get:
P(X=0) = C(10, 0) * (0.5^0) * ((1-0.5)^(10-0))
Calculating this will give you the probability that none of the 10 randomly selected freshmen graduates from the local community college.
Let's calculate:
C(10, 0) = 1 (since there is only one way to choose 0 items from 10)
(0.5^0) = 1 (since any number to the power of 0 is 1)
((1-0.5)^(10-0)) = (0.5^10) = 0.0009765625
So, P(X=0) = 1 * 1 * 0.0009765625 = 0.0009765625
Rounding to four decimal places, the probability that none of the 10 randomly selected freshmen graduates from the local community college is 0.0009765625, or 0.00098.
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