A manufacturer of computer disks has a historical defective rate of .001. What is the probability that in a batch oof 1000 disks, 2 would be defective?
To calculate the probability that in a batch of 1000 disks, 2 would be defective, you can use the binomial distribution formula. The binomial distribution is commonly used to model situations where there are two possible outcomes (success or failure) and a fixed number of independent trials.
The formula for the probability mass function of the binomial distribution is:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting exactly k successes,
- n is the total number of trials,
- k is the number of successes,
- p is the probability of success for each trial, and
- C(n, k) is the number of combinations of n items taken k at a time.
In this case:
- The total number of trials (n) is 1000.
- The number of successes (k) is 2.
- The probability of success for each trial (p) is 0.001 (defective rate).
To determine the probability, substitute these values into the formula:
P(X=2) = C(1000, 2) * (0.001)^2 * (1 - 0.001)^(1000 - 2)
Now let's calculate each part individually:
C(1000, 2) = 1000! / (2! * (1000-2)!)
= 1000! / (2! * 998!)
= (1000 * 999) / (2 * 1)
= 499500
(0.001)^2 = 0.000001
(1 - 0.001)^(1000 - 2) = 0.999^998 ≈ 0.818
Now substitute these values back into the original equation:
P(X=2) = 499500 * 0.000001 * 0.818
P(X=2) ≈ 0.000409
Therefore, the probability that in a batch of 1000 disks, 2 would be defective is approximately 0.000409.