a machine has 0.04 probability of producing a defective item. If it produces 6 items find:
a. probability none is defective
b. at least one is defective
c. all are defective
Use a binomial probability formula or use a binomial probability table.
Formula:
P(x) = (nCx)(p^x)[q^(n-x)]
For a): Find P(0) for none
For b): Take 1 - P(0) for at least 1
For c): Find P(6) for all
If you use the table, p = .04, n = 6 (sample size), x = the values needed for a, b, c.
(Note: q in the formula is 1-p).
I hope this will help get you started.
0.76
To solve these problems, we can use the concept of binomial probability.
a. Probability that none is defective:
The probability of a single item being non-defective is 1 - 0.04 = 0.96.
Since the probability of each item being non-defective is the same, we can use the binomial probability formula: P(X=k) = nCk * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and nCk is the binomial coefficient.
In this case, n = 6 (6 items), k = 0 (none defective), and p = 0.96.
P(X=0) = 6C0 * 0.96^0 * (1-0.96)^(6-0)
= 1 * 1 * 0.04^6
= 0.04^6
= 0.000001536
Therefore, the probability that none of the 6 items is defective is approximately 0.000001536.
b. Probability that at least one is defective:
To find this probability, we will calculate the complement of the probability that none is defective. The complement of an event A is the probability of A not occurring.
P(at least one defective) = 1 - P(none defective)
= 1 - 0.000001536 (from part a)
= 0.999998464
Therefore, the probability that at least one of the 6 items is defective is approximately 0.999998464.
c. Probability that all are defective:
The probability of a single item being defective is 0.04, and since each item is independent, we can multiply the probabilities together.
P(all defective) = 0.04^6
= 0.000001536 (same as from part a)
Therefore, the probability that all 6 items are defective is approximately 0.000001536.
To find the probabilities, we can use the binomial distribution formula:
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 number of trials
- k is the number of successes
- p is the probability of success in a single trial
- (1-p) is the probability of failure in a single trial
- C(n, k) is the binomial coefficient, which represents the number of ways to choose k items from n items without regard to order and is calculated as C(n, k) = n! / (k! * (n-k)!)
Now let's answer the specific questions:
a. Probability that none is defective:
Since the probability of producing a defective item is 0.04, the probability of producing a non-defective item is 1 - 0.04 = 0.96.
Using the formula, we can calculate:
P(X=0) = C(6, 0) * 0.96^0 * 0.04^(6-0) = 1 * 1 * 0.96^0 * 0.04^6 = 0.96^0 * 0.04^6 = 1 * 0.0004 = 0.0004
Therefore, the probability that none of the six items is defective is 0.0004.
b. Probability that at least one is defective:
To find the probability of at least one defective item, we need to calculate the probability of getting 1, 2, 3, 4, 5, or 6 defective items, and then sum up those probabilities.
Using the complement rule, we can calculate the probability of no defective items and subtract it from 1:
P(at least one defective) = 1 - P(none defective)
We already calculated P(none defective) as 0.0004, so:
P(at least one defective) = 1 - 0.0004 = 0.9996
Therefore, the probability that at least one of the six items is defective is 0.9996.
c. Probability that all are defective:
Since the probability of producing a defective item is 0.04, the probability of producing all defective items is:
P(all defective) = P(X=6) = C(6, 6) * 0.04^6 * (1-0.04)^(6-6) = 1 * 0.04^6 * 0.96^0 = 0.04^6 = 0.0000064
Therefore, the probability that all six items are defective is 0.0000064.