90% of the world’s large corporations are actively involved in data warehousing. In a random sample of 10 large corporations, what is the probability that at least 8 of them are actively involved in data warehousing

To find the probability that at least 8 out of 10 large corporations are actively involved in data warehousing, we can use the binomial probability formula.

The binomial probability formula is given by:

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

Where:
- P(X = k) is the probability of getting exactly k successes
- n is the number of trials/experiments
- k is the number of successes
- p is the probability of success on a single trial
- (nCk) is the binomial coefficient, also known as "n choose k"

In this case, n = 10 (number of large corporations in the sample), p = 0.9 (probability that a large corporation is actively involved in data warehousing), and we want to find the probability of getting at least 8 successes (k >= 8).

So, we need to calculate the probabilities for k = 8, 9, and 10, and then add them together.

P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10)

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

Calculating each term:

P(X = 8) = (10C8) * 0.9^8 * (1-0.9)^(10-8)
P(X = 9) = (10C9) * 0.9^9 * (1-0.9)^(10-9)
P(X = 10) = (10C10) * 0.9^10 * (1-0.9)^(10-10)

Next, we need to calculate the binomial coefficients:

(10C8) = 10! / (8! * (10-8)!)
(10C9) = 10! / (9! * (10-9)!)
(10C10) = 10! / (10! * (10-10)!)

Calculating the probabilities:

P(X = 8) = (45) * (0.9)^8 * (0.1)^2
P(X = 9) = (10) * (0.9)^9 * (0.1)^1
P(X = 10) = (1) * (0.9)^10 * (0.1)^0

Finally, we can sum up the probabilities:

P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10)

Once you fill in the values and perform the calculations, you should get the probability that at least 8 out of 10 large corporations are actively involved in data warehousing.