Subject uses drug
44 (True Positive), 6(False Negative)
Subject is not a drug user
90(False Positive), 860(True Negative)
If 2 of the 1000 test subjects are randomly selected, find the probability that they all had false negative results. Is it unlikely to randomly select 3 subjects and get 3 results that are all false negative results?
a. Assume that the 3 selections are made with replacement.
b. Assume that the 3 selections are made without replacement.
a. To find the probability that all three selected subjects have false negative results, we need to multiply the probabilities of each individual selection having a false negative result.
Given that there are 1000 test subjects, with 6 false negative results, the probability of selecting a subject with a false negative result is 6/1000 = 0.006.
Since the selections are made with replacement, the probability of all three selections having false negative results is (0.006)^3 = 0.000000216.
Therefore, the probability of randomly selecting three subjects with all false negative results is 0.000000216 or 0.0000216%.
b. If the selections are made without replacement, the probability will change after each selection because the pool of subjects is decreasing.
For the first selection, the probability of getting a subject with a false negative result is still 6/1000 = 0.006.
However, for the second selection, there are now only 999 subjects left, with 5 false negative results remaining. Therefore, the probability of the second selection having a false negative result is 5/999 ≈ 0.005005005.
For the third selection, there are now only 998 subjects left, with 4 false negative results remaining. Therefore, the probability of the third selection having a false negative result is 4/998 ≈ 0.004008016.
To find the probability of all three selections having false negative results, we need to multiply the probabilities of each individual selection having a false negative result: (0.006) * (0.005005005) * (0.004008016) ≈ 0.000000120.
Therefore, the probability of randomly selecting three subjects without replacement and getting three results that are all false negative results is 0.000000120 or 0.000012%.
To find the probability of getting three false negative results, we need to consider both cases: (a) with replacement and (b) without replacement.
(a) With Replacement:
If the selections are made with replacement, it means that after each selection, the subject is put back into the pool, so they can be selected again. Since we have a total of 1000 subjects, the probability of selecting a false negative subject is 6/1000.
To find the probability of selecting three false negative subjects with replacement, we need to multiply this probability by itself three times because the selections are independent events.
P(3 false negatives with replacement) = (6/1000)^3
(b) Without Replacement:
If the selections are made without replacement, it means that after a subject is selected, it is not put back into the pool. Therefore, the probability of selecting a false negative subject changes with each selection.
For the first selection, the probability of selecting a false negative subject is 6/1000.
For the second selection, after one false negative has already been selected, there are now only 999 subjects remaining, out of which 5 are false negatives. So the probability of selecting a second false negative subject is 5/999.
Similarly, for the third selection, after two false negatives have already been selected, there are now only 998 subjects remaining, out of which 4 are false negatives, so the probability is 4/998.
To find the probability of selecting three false negative subjects without replacement, we need to multiply these probabilities together.
P(3 false negatives without replacement) = (6/1000) * (5/999) * (4/998)
Now, to determine if it is unlikely to randomly select three subjects and get three results that are all false negative results, we need to compare these probabilities to a threshold. The threshold value is usually arbitrary and depends on the context or requirements of the problem. In general, a probability less than 0.05 (5%) is often considered unlikely.
It is important to note that these calculations are based on the given information, assuming that the test's accuracy remains constant over the selection process.