The probability that a certain production process will produce a defective part is 0.20. If a box contains 12 parts, the probability that it will contain 6 defective parts is?
To find the probability that a box contains 6 defective parts out of 12, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting k successes (in this case, getting k defective parts)
C(n, k) is the number of ways to choose k objects from n objects (also known as the binomial coefficient)
p is the probability of getting a success (in this case, getting a defective part)
q = 1 - p is the probability of not getting a success (in this case, getting a non-defective part)
n is the total number of trials (in this case, the number of parts in the box)
In this case, we want to find the probability of getting 6 defective parts out of 12, so we have:
P(X = 6) = C(12, 6) * (0.20)^6 * (1 - 0.20)^(12 - 6)
To calculate this, we need to calculate the binomial coefficient C(12, 6), which represents the number of ways to choose 6 objects out of 12. The binomial coefficient is calculated using the formula:
C(n, k) = n! / (k!(n - k)!)
Where ! represents the factorial operation.
Plugging in the values, we have:
C(12, 6) = 12! / (6!(12 - 6)!) = (12 * 11 * 10 * 9 * 8 * 7) / (6 * 5 * 4 * 3 * 2 * 1) = 924
Substituting this value back into the binomial probability formula, we have:
P(X = 6) = 924 * (0.20)^6 * (1 - 0.20)^(12 - 6)
Evaluating this expression gives us the desired probability.