A manufacturing machine has a 9% defect rate.
If 10 items are chosen at random, what is the probability that at least one will have a defect
prob(good) = .91
prob(not good) = .09
prob(at least defective)
= 1 - prob(all good)
= 1 - .91^10 = .....
Well, I guess you could say that manufacturing machine has a bit of a "defective personality"! But let's calculate the probability of at least one defective item out of 10.
To find the probability, we can use the concept of complementary events. The probability of at least one defective item is equal to 1 minus the probability of no defective items.
The probability of no defective items is calculated by multiplying the probabilities of each item being non-defective. Since the defect rate is 9%, the non-defective rate is 91% or 0.91.
So, for one item being non-defective, the probability is 0.91. The probability of all 10 items being non-defective is obtained by raising 0.91 to the power of 10.
Calculating it, we get:
Probability of no defective items = 0.91^10 = 0.3874
Finally, the probability of at least one defective item is:
Probability of at least one defective item = 1 - Probability of no defective items
= 1 - 0.3874
= 0.6126
So, there's approximately a 61.26% chance of at least one item being defective out of the 10 chosen at random.
To find the probability that at least one of the randomly chosen 10 items will have a defect, we can use the complement rule.
The complement rule states that the probability of an event not occurring is equal to 1 minus the probability of the event occurring.
Here, we want to find the probability that none of the 10 randomly chosen items have a defect. We can calculate this using the binomial probability 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 (0 defects in our case),
- C(n, k) is the number of combinations of choosing k items out of n items,
- p is the probability of getting a single success (defect),
- n is the total number of trials (items chosen).
Using these values, we can calculate the probability of none of the 10 items having a defect:
P(X = 0) = C(10, 0) * 0.09^0 * (1 - 0.09)^(10 - 0)
Simplifying this expression:
P(X = 0) = 1 * 1 * (0.91^10)
P(X = 0) = 0.91^10
Calculating this value:
P(X = 0) ≈ 0.389
Now, to find the probability that at least one item will have a defect, we subtract this probability from 1:
P(at least one defect) = 1 - P(X = 0)
P(at least one defect) = 1 - 0.389
P(at least one defect) ≈ 0.611
Therefore, the probability that at least one item will have a defect is approximately 0.611, or 61.1%.
To calculate the probability of at least one defective item, we need to find the probability that none of the 10 items will have a defect and then subtract this probability from 1.
Let's break down the steps to calculate the probability:
Step 1: Calculate the probability of an item being defective:
Given that the machine has a 9% defect rate, we can say that the probability of an item being defective is 0.09 or 9% (expressed as a decimal).
Step 2: Calculate the probability of an item not being defective:
To find the probability of an item not being defective, we subtract the probability of being defective from 1.
The probability of an item not being defective is therefore 1 - 0.09 = 0.91 or 91%.
Step 3: Calculate the probability that none of the 10 items will have a defect:
Since the items are chosen at random, we can assume that their probabilities are independent. Therefore, we can multiply the probabilities together.
The probability that none of the 10 items will have a defect is (0.91)^10 = 0.3874 (rounded to four decimal places).
Step 4: Calculate the probability of at least one defective item:
To find the probability of at least one defective item, we subtract the probability of none of the items being defective from 1.
The probability of at least one defective item is 1 - 0.3874 = 0.6126 or 61.26% (rounded to two decimal places).
Therefore, the probability of at least one item having a defect when 10 items are chosen at random is 0.6126 or 61.26%.