A sample of four different calculators is randomly selected from a group containing 15 that are defective and 26 that have no defects. Find the probability that at least one of the calculators is defective
P(no defective in 4 draws is) = 26/41 * 25/40 * 24/39 * 23/38 = 115/779
So, P(at least 1) = 1 - P(none) = 664/779
To find the probability that at least one of the calculators is defective when a sample of four calculators is randomly selected, we can use the complement rule.
The complement rule states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring. In this case, the event we are interested in is none of the calculators being defective.
To calculate the probability of none of the calculators being defective, we have to select four calculators that are not defective from the total pool of calculators.
The probability of selecting a non-defective calculator on the first draw is (26 non-defective calculators / total calculators). After each selection, the number of non-defective calculators and the total number of calculators decrease by 1.
Therefore, the probability of selecting four non-defective calculators in a row is:
(26/41) * (25/40) * (24/39) * (23/38)
Now, we can find the probability of at least one calculator being defective by subtracting this probability from 1:
1 - [(26/41) * (25/40) * (24/39) * (23/38)]
Calculating this expression will give us the final answer.
To find the probability that at least one of the calculators is defective, we can use the concept of complementary probability.
First, let's find the probability that none of the calculators is defective. We need to select four calculators out of the total of 26 calculators with no defects. This can be calculated using the combination formula (nCr):
Number of ways to select 4 calculators with no defects = C(26, 4)
Next, let's find the probability that all the calculators are working correctly. There are no defective calculators in this case, so we need to select four calculators out of the total of 26 calculators with no defects:
Number of ways to select 4 calculators with no defects = C(26, 4)
Now, the probability that at least one of the calculators is defective is equal to 1 minus the probability that none of the calculators is defective or the probability that all the calculators are working correctly. Therefore:
Probability(At least one defective calculator) = 1 - (Probability(None defective) + Probability(All working))
Probability(At least one defective calculator) = 1 - [C(26, 4) / C(41, 4) + C(26, 4) / C(41, 4)]
Simplifying this expression will give you the final probability.