a manufacturing machine has a 8% defect rate. if 5 items are chosen at random, what is the probability that at least one will have a defect
Ah, the thrill of probability and manufacturing mishaps! Let's calculate it, shall we?
To find the probability of at least one defect, we can use the concept of complementary events. We'll first calculate the probability of having zero defects in 5 items and then subtract it from 1.
The probability of having no defect in a single item is 1 - 0.08 (defect rate), which equals 0.92.
Now, since we have 5 items chosen at random, the probability of none having defects is (0.92)^5 = 0.6634.
Since we want the opposite event (at least one defect), we'll subtract this probability from 1:
1 - 0.6634 = 0.3366.
So, the probability of having at least one defective item is approximately 0.3366, or 33.66%.
Now, I hope your manufacturing machines don't get too carried away with their love for defects!
To find the probability that at least one item will have a defect, we can use the complement rule. This is the probability that none of the items have a defect subtracted from 1.
Let's calculate the probability:
Step 1: Calculate the probability that an item does not have a defect.
- The defect rate is given as 8% or 0.08
- So, the probability that an item does not have a defect is 1 - 0.08 = 0.92
Step 2: Calculate the probability that none of the 5 items have a defect.
- Since the items are chosen at random and independent, we can calculate this by multiplying the probability that each item does not have a defect.
- Probability of no defect in one item = 0.92
- To find no defect in all 5 items, we multiply this probability 5 times: 0.92^5 = 0.6634
Step 3: Calculate the probability that at least one item will have a defect.
- This is the complement of the probability of no defect in all 5 items.
- So, P(at least one defect) = 1 - 0.6634 = 0.3366
Therefore, the probability that at least one item will have a defect is approximately 0.3366 or 33.66%.
To find the probability that at least one item will have a defect, we need to calculate the probability of the compliment event and subtract it from 1.
Let's break down the problem step by step:
Step 1: Calculate the probability of an item having no defect.
To find this probability, subtract the defect rate from 100%:
Probability of no defect = 100% - 8% = 92% = 0.92
Step 2: Calculate the probability that all 5 items have no defects.
Since the items are chosen randomly and independently, we can multiply the probabilities:
Probability of all 5 items having no defects = 0.92 × 0.92 × 0.92 × 0.92 × 0.92 = 0.66 (approximately)
Step 3: Calculate the probability of at least one item having a defect.
To find this probability, subtract the probability of all items having no defects from 1:
Probability of at least one item having a defect = 1 - 0.66 = 0.34 (approximately)
Therefore, the probability that at least one item will have a defect is approximately 0.34, or 34%.
find P of 5 successes (100 %)
then subtract that from 1
p of success = .92
P of 5 in a row = .92^5 = .659
so probability that not all 5 work
= 1 - .659 = .341