In a batch of 8,000 clock radios 5% are defective. A sample of 14 clock radios is randomly selected without replacement from the 8,000 and tested. The entire batch will be rejected if at least one of those tested is defective. What is the probability that the entire batch will be rejected?
prob defective = .05
prob of good = .95
So we want the prob that all are good
= .95^14
= .48776...
prob of at least one is defective
= 1 - .48776..
= appr. .512
The 8000 does not enter the picture, since we have the percentage
if we used it then we could proceed as follow
numbers defective = .05(8000) = 400
number not defective = 7600
prob not at least one of 14 is defective
= 1 - prob of none defective
= 1 - (7600/8000)^14
= .512
That's the probability of those non-defective. Defective is 0.5 (5%). So 1-0.5=0.95. @Kojo
First one 3/20
Second one 2/19
third one 1/18
3/20*2/19*1/18 = 1/(3*20*19) =1/1140
Well, the probability of selecting a defective clock radio from the batch is 5%, so the probability of not selecting a defective clock radio is 95%.
Since we are sampling without replacement, the probability of not selecting a defective radio for the first test is 95%. For the second test, since one clock radio has already been removed, the probability of not selecting a defective radio is 94.74%. And so on, until we reach the 14th test, where the probability will be 92.18%.
To find the probability that the entire batch will be rejected, we need to calculate the product of these probabilities for all 14 tests:
P(rejecting the entire batch) = 0.95 × 0.9474 × 0.9449 × ... × 0.9218
Calculating this product, we find that the probability is approximately 40.53%. So, there's a 40.53% chance of the entire batch being rejected.
Looks like these clock radios have a faulty sense of timing!
To find the probability that the entire batch will be rejected, we need to find the probability that at least one of the 14 clock radios tested is defective.
Let's break down the problem step by step:
Step 1: Find the probability that a randomly selected clock radio from the batch is defective. Given that 5% of the 8,000 clock radios are defective, we can find the probability as follows:
Probability of a defective clock radio = (Number of defective clock radios) / (Total number of clock radios)
Probability of a defective clock radio = 0.05
Step 2: Find the probability that none of the 14 clock radios tested are defective. Since we are sampling without replacement, the probability changes for each radio tested. We can calculate this by multiplying the probabilities for each tested radio.
Probability of none of the 14 tested clock radios being defective = (Prob of the 1st radio being non-defective) * (Prob of the 2nd radio being non-defective) * ... * (Prob of the 14th radio being non-defective)
For the first radio, the probability of it being non-defective is 1 - 0.05 (as we calculated in Step 1). Since we are sampling without replacement, for the second radio, the probability of it being non-defective is now (Number of non-defective radios) / (Total number of remaining radios). The number of non-defective radios is 8,000 - 1 (the first radio tested), and the total number of remaining radios is 8,000 - 1 (the first radio tested) - 1 (the second radio tested). So the probability of the second radio being non-defective is (8,000 - 1) / (8,000 - 1 - 1).
Applying this logic, we can calculate the probability of none of the 14 tested radios being defective.
Step 3: Calculate the complement of the probability in Step 2. In other words, if the probability of none of the tested radios being defective is p, then the probability of at least one of them being defective is 1 - p.
Following these steps, we can calculate the probability that the entire batch will be rejected.
Note: The calculations involve high precision and can be complex. To simplify the process, you can use software or a statistical calculator.