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A company produces shotgun shells in batches of 350. A sample of 5 is tested from each batch, and if more than one defect is found, the entire batch is tested. (Round your answers to five decimal places.) (a) If 1% of the shells are actually defective and we assume independence, what is the probability of 0 defective shells in the sample?
(b) If 1% of the shells are actually defective and we assume independence, what is the probability of 1 defective shell in the sample?
(c) If 1% of the shells are actually defective and we assume independence, what is the probability of more than 1 defective shell in the sample?

Sagot :

a. The probability of defective shells from the given samples are:

a. Probability of zero defective shells of samples is equal to 0.9510.

b. Probability of one defective shells of samples is equal to 0.04803

c. Probability of more than 1 defective shells of samples is equal to 0.00107.

As given in the question,

Total number of batches = 350

Sample size used to test for each batch 'n' = 5

Actual defective shells (success) 'p' = 1%

                                                           = 0.01

Actual non-defective shells ' 1 - p ' = 1 - 0.01

                                                         = 0.99

a. Probability of zero defective shells from the given samples is

=  ⁵C₀ × ( 0.01 )⁰ × (0.99)⁵⁻⁰

= [ (5!)/(5-0)!0! ] × 1 × ( 0.99)⁵

= 0.9510

b. Probability of 1 defective shells from the given samples is

= ⁵C₁ × ( 0.01 )¹ × (0.99)⁵⁻¹

= [ (5!)/(5-1)!1! ] × 0.01 × ( 0.99)⁴

= 5 × 0.01 × 0.9606

=0.04803

c. Probability of more than 1 defective shells from the given samples is

= P(2) + P(3) + P(4) + P(5)

= ⁵C₂ × ( 0.01 )²× (0.99)⁵⁻² + ⁵C₃ × ( 0.01 )³ × (0.99)⁵⁻³ + ⁵C₄× ( 0.01 )⁴ × (0.99)⁵⁻⁴+ ⁵C₅× ( 0.01 )⁵× (0.99)⁵⁻⁵

= 0.00097 + 0.000098 +0.000000044 +0.0000000001

=0.00107

Therefore, the probability of following conditions are as follow:

a. Probability of zero defective shells of samples is equal to 0.9510.

b. Probability of one defective shells of samples is equal to 0.04803

c. Probability of more than 1 defective shells of samples is equal to 0.00107.

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