Final answer:
The probability that exactly four out of 20 contracts sampled will experience cost overruns is calculated using the binomial distribution formula, resulting in approximately 13.53% chance.
Step-by-step explanation:
The probability that exactly four of the 20 contracts experience cost overruns is a question that can be answered using the binomial distribution, where the number of successes in a sequence of n independent experiments is counted. The binomial probability formula is given by P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n' is the number of trials, 'k' is the number of successful trials, 'p' is the probability of success on an individual trial, and '(1-p)' is the probability of failure on an individual trial.
Given that the probability of cost overruns on a contract is 0.28, and 20 contracts are sampled, the probability of exactly four cost overruns can be calculated as follows:
P(X = 4) = (20 choose 4) * (0.28)^4 * (1-0.28)^(20-4).
Using the formula, we calculate:
(20 choose 4) = 20! / (4!(20-4)!) = 4845,
(0.28)^4 = 0.0061504,
(1-0.28)^(20-4) = (0.72)^16 = 0.043102,
Thus, P(X = 4) = 4845 * 0.0061504 * 0.043102 = 0.135264, which is approximately 13.53%.
The student can interpret this as there being a 13.53% chance that exactly four out of the 20 contracts sampled will experience cost overruns.