1. A local community college determines the probability that a student will reenroll for a second year is 0.91. A representative surveys 100 random first year students and asks them if they will be enrolling for the next year. What is the probability that exactly 91 of them will enroll?

a. 0.138
b. 1.000
c. 0.910 <--my answer
d. 0.276

C(100,91) (.91)^91 (.09)^9

= .138106...

The hard part was to evaluate C(100,91)
If you use the build-in function n C r ,
your calculator will overload and you get an error message.
that is because the largest number most calculators can store is 96!
so you have to do C(100,91)
= 100x99x98x97x96x95x94x93x92/9!

I meant to say:

that is because the largest number most calculators can store is 69!

This had no effect on my conclusion or answer.

To find the probability that exactly 91 of the 100 random first-year students will enroll for the next year, we can use the binomial probability formula.

The binomial probability formula is:

P(X=k) = (nCk) * p^k * (1-p)^(n-k)

Where:
P(X=k) is the probability of getting exactly k successes (students who will enroll),
n is the total number of trials (100 students in this case),
k is the number of successful outcomes (91 students in this case),
p is the probability of success in a single trial (0.91 in this case), and
1-p is the probability of failure in a single trial.

Using the binomial probability formula, we can calculate:

P(X=91) = (100C91) * (0.91)^91 * (1-0.91)^(100-91)

To calculate (100C91), which represents the number of ways to choose 91 students out of 100, we can use the combination formula:

(100C91) = 100! / (91!(100-91)!)

Calculating the above terms, we get:

(100C91) = 100! / (91! * 9!) = 100 * 99 * 98 * 97 * 96 * 95 * 94 * 93 * 92 / (91 * 90 * 89 * 88 * 87 * 86 * 85 * 84 * 83)= 42,188,805,080

Now, we can substitute the values back into the binomial probability formula:

P(X=91) = (42,188,805,080) * (0.91)^91 * (1-0.91)^(100-91)

Calculating further:

P(X=91) ≈ 0.138

Therefore, the correct answer is option a) 0.138.

To find the probability that exactly 91 of the 100 randomly surveyed first-year students will enroll for the next year, we can use the binomial probability formula.

The binomial probability formula is given as:

P(X = k) = (n C k) * p^k * (1-p)^(n-k),

where P(X = k) is the probability of exactly k successes (in this case, exactly 91 students enrolling), n is the total number of trials (in this case, 100 students surveyed), p is the probability of success for each trial (in this case, 0.91), and (n C k) represents the number of combinations of n things taken k at a time.

Applying the values to the formula, we have:

P(X = 91) = (100 C 91) * 0.91^91 * (1-0.91)^(100-91)

To calculate (100 C 91), which represents the number of combinations of 100 things taken 91 at a time, we can use the formula:

(100 C 91) = 100! / (91! * (100-91)!)

Simplifying this expression, we get:

(100 C 91) = 100! / (91! * 9!)

Now, we can calculate the probability:

P(X = 91) = (100! / (91! * 9!)) * 0.91^91 * (1-0.91)^(100-91)

Using a calculator or statistical software, we can find that P(X = 91) ≈ 0.138.

Therefore, the correct answer is option a. 0.138.