2. In a certain state, it has been shown that only 56% of the high school graduates who are capable of college work actually enroll in colleges.Find the probability that, among the 11 capable high school graduates in this state, each number will enroll in college

at least 4

11 * 0.56 = 6.26 = 6

To find the probability that among the 11 capable high school graduates in this state, each number will enroll in college at least 4, you can use the binomial probability formula.

The binomial probability formula is given by:
P(x) = (nCx) * p^x * q^(n-x)

Where:
P(x) is the probability of getting exactly x successes.
n is the number of trials (in this case, the number of high school graduates).
x is the number of successes (in this case, the number of graduates enrolling in college).
p is the probability of success in a single trial (in this case, the probability of a high school graduate enrolling in college).
q is the probability of failure in a single trial (in this case, the probability of a high school graduate not enrolling in college).
nCx is the binomial coefficient, which represents the number of ways to choose x successes from n trials.

In this case, we want to find the probability that each of the 11 capable high school graduates enrolls in college at least 4 times. The probability of a single graduate enrolling in college is 56% or 0.56, so p = 0.56. The probability of a single graduate not enrolling in college is 1 - p = 1 - 0.56 = 0.44, so q = 0.44.

Using the binomial probability formula, we can calculate the probability of each graduate enrolling in college at least 4 times by summing the individual probabilities for each x from 4 to 11.

P(at least 4 enrollments) = P(x=4) + P(x=5) + P(x=6) + P(x=7) + P(x=8) + P(x=9) + P(x=10) + P(x=11)

Substituting the values into the formula, we get:

P(at least 4 enrollments) = (11C4 * 0.56^4 * 0.44^7) + (11C5 * 0.56^5 * 0.44^6) + (11C6 * 0.56^6 * 0.44^5) + (11C7 * 0.56^7 * 0.44^4) + (11C8 * 0.56^8 * 0.44^3) + (11C9 * 0.56^9 * 0.44^2) + (11C10 * 0.56^10 * 0.44^1) + (11C11 * 0.56^11 * 0.44^0)

You can calculate each term using a binomial coefficient table or a calculator that supports combinatorial calculations. Once you have the values, simply add them up to find the probability.