Answer:
(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
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
Hope this Helps!!!