A manufacturing machine has a 10% defect rate.
If 10 items are chosen at random, what is the probability that at least one will have a defect?
To calculate the probability that at least one item will have a defect, we can use the concept of complementary probability. The complementary probability states that the probability of an event not occurring is equal to 1 minus the probability of the event occurring.
In this case, we want to find the probability that at least one item will have a defect. This means we want to find the probability of the event "no defects" not happening.
Step 1: Calculate the probability of no defects occurring:
The probability of choosing an item without a defect is 1 minus the defect rate, which in this case is 10%. So, the probability of choosing an item without a defect is 1 - 0.10 = 0.90.
Step 2: Calculate the probability of no defects in 10 items:
Since we are choosing 10 items at random, the probability of no defects occurring in any one item is 0.90. Assuming that each item is selected independently, we can multiply the probabilities together to get the probability of no defects in all 10 items.
So, the probability of no defects in 10 items is 0.90^10.
Step 3: Calculate the probability of at least one defect:
Finally, we can calculate the probability of at least one defect by subtracting the probability of no defects from 1.
So, the probability of at least one defect is 1 - 0.90^10.
Using a calculator, you can evaluate this expression to find the numerical value of the probability.