Question

in a quality control procedure, the chance that a mass-produced article will fail to pass a guage is 0.16 . calculate, to 2 decimal places, the chance that more than of 2 the sample of 10 articles will fail to pass.

Answer 1

To calculate the probability of more than 2 out of 10 articles failing, we use the binomial probability formula, considering the chance of individual failure as 0.16, and subtract the probabilities of 0, 1, and 2 failures from 1.

Step-by-step explanation:

The task involves calculating the probability of more than 2 out of 10 articles failing a quality control check, given that the chance of an individual article failing is 0.16. To solve this, we can use the binomial probability formula:

P(X > k) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]

Where:

P(X > k) is the probability of more than k failures.

X follows a binomial distribution with parameters n (number of trials) and p (probability of failure).

First, we calculate the probabilities of exactly 0, 1, and 2 failures:

P(X = 0) = C(n, 0) * (p)^0 * (1-p)^(n-0)

P(X = 1) = C(n, 1) * (p)^1 * (1-p)^(n-1)

P(X = 2) = C(n, 2) * (p)^2 * (1-p)^(n-2)

Next, we subtract the sum of these probabilities from 1 to get the probability of more than 2 failures.

Calculations are done using the binomial coefficients for combinations C(n, k), and given p = 0.16 and n = 10:

C(10, 0) = 1, C(10, 1) = 10, C(10, 2) = 45

p^0 = 1, p^1 = 0.16, p^2 = 0.0256

(1-p)^10 ≈ 0.1820, (1-p)^9 ≈ 0.2693, (1-p)^8 ≈ 0.3763

Thus we compute P(X = 0), P(X = 1), and P(X = 2) using the binomial probability formula for each, add them up, and subtract from 1 to get the final probability, rounding to two decimal places as requested.