A sample of 4 different calculators is randomly selected from a group containing 10 that are defective and 35 that have no defects. What is the probability that at least one of the calculators is defective?
To find the probability that at least one calculator is defective, we can use the concept of complementary probability, which states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring.
First, let's find the probability that none of the calculators are defective. To calculate this, we need to find the probability of selecting 4 non-defective calculators from the total 45 calculators (10 defective calculators + 35 non-defective calculators).
The probability of selecting a non-defective calculator in the first pick is 35/45 (since there are 35 non-defective calculators out of 45 total).
Once a non-defective calculator has been selected, the probability of selecting another non-defective calculator in the second pick becomes 34/44 (as there will be 34 non-defective calculators remaining out of 44 total calculators).
This process continues for the subsequent picks. So, the probability of selecting 4 non-defective calculators is:
(35/45) * (34/44) * (33/43) * (32/42) ≈ 0.377
Now, to find the probability that at least one calculator is defective, we subtract the probability of none of the calculators being defective from 1:
1 - 0.377 = 0.623
Therefore, the probability that at least one calculator is defective is approximately 0.623, or 62.3%.