Question

Each of 12 refrigerators of a certain type has been returned to a distributorbecause of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that 7 of these refrigerators have a defective compressor andthe other 5 have less serious problems. If the refrigerators are examined inrandom order, let X be the number among the first 6 examined that have adefective compressor. Answer the following questions.
(a) P (X = 5)
(b) P (X <_ 4)

Answer 1

Answer:

a.) 0.1685

b.) 0.7922

Step-by-step explanation:

This question is solved by the probability distribution formula for selection which uses the combination formula to derive the answer.

If 7 out of 12 cases Of refrigerators are said to have compressor problem, then the success rate to be used in the probability distribution formula = p = 7/12 = 0.583

Probability of failure = q = 0.417

When choosing r samples from a possible n sample, then probability distribution formula is denoted by :

P(X=r) = nCr × p^n × q^n-r

a.) From possible 6 samples, if P(X=5)

P(X=5) = 6C5 × 0.583^5 × 0.417¹

P(X=5) = 0.1685

b.) P(X≤4) =1 -P(X>4) = 1 - P(X=5) - P(X=6)

P(X=5) as calculated earlier = 0.1685

P(X=6) = 6C6 × 0.583^6 × 0.417^0

P(X=6) = 0.0393.

Hence, P(X≤4) = 1- 0.1685 - 0.0393

P(X≤4) = 0.7922