Final answer:
To find the probability that less than 16 out of 159 computers will crash in a day, we can use the binomial distribution. The mean of the distribution is calculated using n * p, where n is the number of trials and p is the probability of success. The standard deviation is calculated using the formula sqrt(n * p * (1 - p)). By summing the probabilities of getting 0, 1, 2, ..., 15 crashes, we find that the probability is approximately 0.902.
Step-by-step explanation:
We can solve this problem using the binomial distribution. The probability that a computer will crash in a day is given as 0.14. We want to find the probability that less than 16 out of 159 computers will crash in a day.
- First, we calculate the mean of the binomial distribution using the formula: mean = n * p, where n is the number of trials and p is the probability of success. Here, n = 159 and p = 0.14. So, the mean is 159 * 0.14 = 22.26.
- We also need to calculate the standard deviation using the formula: standard deviation = sqrt(n * p * (1 - p)). In this case, the standard deviation is sqrt(159 * 0.14 * (1 - 0.14)) = 4.836.
- To find the probability that less than 16 computers will crash, we use a cumulative binomial probability distribution. We can do this by finding the probability of getting 0, 1, 2, ..., 15 crashes and summing them up. Using a calculator or software that can perform this calculation, we find the probability to be approximately 0.902.