A quality-control inspector rejects any shipment of printed circuit boards whenever 3 or more defectives are found in a sample of 50 boards tested. Find the (1) expected number defective and (2) the probability of rejecting the shipment when the proportion of defectives in the entire shipment is

(a) Proportion of Defectives=0.01

Expected Number Defective=

Probability of Rejecting the Shipment=

(b) Proportion of Defectives=0.05

Expected Number Defective=

Probability of Rejecting the Shipment=

(c) Proportion of Defectives=0.1

Expected Number Defective=

Probability of Rejecting the Shipment=

Expert

(a) Proportion of Defectives = 0.01

Expected Number Defective = 0.5

Probability of Rejecting the Shipment = 0.0138

(b) Proportion of Defectives = 0.05

Expected Number Defective = 2.5

Probability of Rejecting the Shipment = 0.459

(c) Proportion of Defectives = 0.1

Expected Number Defective = 5

Probability of Rejecting the Shipment = 0.888

Step-by-step explanation:

This is a binomial distribution problem due to the unchanging probability of getting a defective board, no matter the number of trials ran.

The expected number for binomial distribution is given as E(X) = np

The probability mass funaction for binomial distribution is given as

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = 50

x = Number of successes required = ≥3

p = probability of success = changing from question to question

q = probability of failure = 1 - p

Total number of boards tested = 50

Note that probability of rejecting the shipment for each of the sub-question is the probability that 3 or more boards are defective. That is, P(X ≥ 3)

P(X ≥ 3) = 1 - P(X < 3) = 1 - [P(X=0) + P(X=1) + P(X=2)]

a) Proportion of Defectives=0.01

Expected Number Defective = np = 0.01 × 50 = 0.5

Probability of Rejecting the Shipment = P(X ≥ 3)

n = total number of sample spaces = 50

p = probability of success = probability of a detective board = 0.01

q = probability of failure = 1 - 0.01 = 0.99

P(X ≥ 3) = 1 - P(X < 3) = 1 - [P(X=0) + P(X=1) + P(X=2)] = 0.01381727083 = 0.0138

b) Proportion of Defectives = 0.05

Expected Number Defective = np = 0.05 × 50 = 2.5

Probability of Rejecting the Shipment = P(X ≥ 3)

n = total number of sample spaces = 50

p = probability of success = probability of a detective board = 0.05

q = probability of failure = 1 - 0.05 = 0.95

P(X ≥ 3) = 1 - P(X < 3) = 1 - [P(X=0) + P(X=1) + P(X=2)] = 0.45946687728 = 0.459

c) Proportion of Defectives = 0.1

Expected Number Defective = np = 0.1 × 50 = 5

Probability of Rejecting the Shipment = P(X ≥ 3)

n = total number of sample spaces = 50

p = probability of success = probability of a detective board = 0.1

q = probability of failure = 1 - 0.1 = 0.90

P(X ≥ 3) = 1 - P(X < 3) = 1 - [P(X=0) + P(X=1) + P(X=2)] = 0.88827124366 = 0.888

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