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 distribution is used when there are only two possible outcomes: success or failure. In this case, success would be a large corporation actively involved in data warehousing, and failure would be a large corporation not involved in data warehousing.
The probability of success, denoted by p, can be calculated as the proportion of large corporations actively involved in data warehousing, which is given as 90% or 0.9. The probability of failure, denoted by q, is simply 1 minus the probability of success or 0.1.
The formula for the probability of getting exactly k successes in a random sample of n trials is:
P(k successes) = (n C k) * p^k * q^(n-k)
where (n C k) denotes the combination or binomial coefficient, which can be calculated as n! / (k! * (n-k)!).
To find the probability of at least 8 successes, we need to sum the probabilities of getting 8, 9, and 10 successes.
P(at least 8 successes) = P(8 successes) + P(9 successes) + P(10 successes)
= [(10 C 8) * 0.9^8 * 0.1^2] + [(10 C 9) * 0.9^9 * 0.1^1] + [(10 C 10) * 0.9^10 * 0.1^0]
To calculate the probabilities, you can use a calculator or statistical software, or consult a binomial probability table.
Note: The binomial distribution assumes each trial is independent and has the same probability of success.