Problem 3: A computer laboratory in a school has 40 computers. Each of the 40 computers has 90% reliability; that is, the probability that a given computer works is 90%. Let X be the number of computers that are working.

a. Write few sentences explaining how / why X satisfies the conditions of a binomial random variable.

b. Find the probability that exactly 30 of the computers are working. Write the formula (the binomial probability distribution formula), the computer command, and actually use the computer to find the probability rounded to the nearest thousandth.

c. Find the probability that at most 20 of the computers are working. Write the formula (the binomial probability distribution formula), the computer command, and actually use the computer to find the probability rounded to the nearest thousandth. 3

d. Assume that a class of 35 students is taken into the lab. Find the probability that there are enough working computers for all the students. That is, find the probability that at least 35 of the computers are working. Write the formula (the binomial probability distribution formula), the computer command, and actually use the computer to find the probability rounded to the nearest thousandth.

e. Find the expected value (mean) of X.

f. Find the variance of X.

g. Find the standard deviation of X. Round the result to the nearest hundredth.