Manufacturing and Product Development In a quarter-pound bag of red pistachio nuts, some shells are too difficult to pry open by hand. Suppose the random variable X, the number of pistachios in a...

Manufacturing and Product Development In a quarter-pound bag of red pistachio nuts, some shells are too difficult to pry open by hand. Suppose the random variable X, the number of pistachios in a randomly selected bag that cannot be opened by hand, has the probability distribution given in the table below.

a. Is this a valid probability distribution? Justify your answer.

b. Find the mean, variance, and standard deviation of X.

c. Find the probability X takes on a value less than the mean.

d. Suppose two bags of pistachios are selected at random. What is the probability that both bags have four or more pistachios too difficult to open by hand?