When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let X = the number of defective boards in a random sample of size n = 25, so X ~ Bin(25, 0.05). (Round your probabilities to three decimal places.)
(a) Determine P(X <_ 2).(b) Determine P(X >_ 5).(c) Determine P(1 <_ X <_ 4).(d) What is the probability that none of the 25 boards is defective?

(a) The value of P (X ≤ 2) is 0.873.

(b) The value of P (X ≥ 5) is 0.007.

(c) The value of P (1 ≤ X ≤ 4) is 0.716.

(d) The probability that none of the 25 boards is defective is 0.277.

Step-by-step explanation:

The random variable X is defined as the number of defective circuit boards.

It is provided that X follows a Binomial distribution with parameter n = 25 and p = 0.05.

The probability mass function of Binomial distribution is:

(a)

Compute the value of P (X ≤ 2) as follows:

P (X ≤ 2) = P (X = 0) + P (X = 1) + P (X = 2)

Thus, the value of P (X ≤ 2) is 0.873.

(b)

Compute the value of P (X ≥ 5) as follows:

P (X ≥ 5) = 1 - P (X < 5)

              = 1 - P (X = 0) - P (X = 1) - P (X = 2) - P (X = 3)+- P (X = 4)

Thus, the value of P (X ≥ 5) is 0.007.

(c)

Compute the value of P (1 ≤ X ≤ 4) as follows:

P (1 ≤ X ≤ 4) = P (X = 1) + P (X = 2) + P (X = 3) + P (X = 4)

Thus, the value of P (1 ≤ X ≤ 4) is 0.716.

(d)

Compute the probability of X = 0 as follows:

Thus, the probability that none of the 25 boards is defective is 0.277.