In a manufacturing process, it is estimated that only 2% of the bolts that are machined are declared defective, that is, they are either too large or too small. In a package of 50 bolts, what is the probability that there is at least one defective bolt?
To find the probability of at least one defective bolt in a package of 50, we can use the complement rule. The complement rule states that the probability of an event not happening is equal to one minus the probability of the event happening.
First, let's calculate the probability of a single bolt being defective. We are given that the estimated percentage of defective bolts is 2%, which can be written as 0.02 in decimal form.
Therefore, the probability of a single bolt being defective is 0.02.
Now, we can find the probability of no defective bolts in a package of 50 by multiplying the probability of a single bolt not being defective (1 - 0.02) with itself 50 times since each bolt is independent.
(1 - 0.02)^50 = 0.98^50 ≈ 0.3641
Next, we can use the complement rule to find the probability of at least one defective bolt:
P(at least one defective bolt) = 1 - P(no defective bolts)
= 1 - 0.3641
≈ 0.6359
So, the probability of at least one defective bolt in a package of 50 is approximately 0.6359, or 63.59%.