HELP PLEASE!!!
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
To find the probability that both test subjects had false negative results, we first need to determine the probability of each individual subject having a false negative result.
For the scenario with replacement (a), the probability of having a false negative result is the ratio of false negatives to the total number of subjects:
P(False Negative (with replacement)) = False Negative / Total Subjects
P(False Negative (with replacement)) = 6 / 1000
P(False Negative (with replacement)) = 0.006
Since the selections are made independently with replacement, the probability of both subjects having false negative results is simply the product of their individual probabilities:
P(Both False Negative (with replacement)) = P(False Negative (with replacement)) * P(False Negative (with replacement))
P(Both False Negative (with replacement)) = 0.006 * 0.006
P(Both False Negative (with replacement)) = 0.000036
So the probability of randomly selecting two subjects with false negative results in this scenario is 0.000036 or 0.0036%.
For the scenario without replacement (b), the calculation is a bit different because the selections are made without replacement. The probability of having a false negative result for the first subject remains unchanged:
P(False Negative (without replacement)) = False Negative / Total Subjects
P(False Negative (without replacement)) = 6 / 1000
P(False Negative (without replacement)) = 0.006
However, for the second subject, the total number of subjects decreases by 1 because the first subject has already been selected. Therefore, the probability of having a false negative result for the second subject is:
P(False Negative (without replacement, second subject)) = False Negative / (Total Subjects - 1)
P(False Negative (without replacement, second subject)) = 6 / (1000 - 1)
P(False Negative (without replacement, second subject)) = 6 / 999
To calculate the probability of both subjects having false negative results, we again multiply their individual probabilities:
P(Both False Negative (without replacement)) = P(False Negative (without replacement)) * P(False Negative (without replacement, second subject))
P(Both False Negative (without replacement)) = 0.006 * 6 / 999
P(Both False Negative (without replacement)) = 0.003618619
So the probability of randomly selecting two subjects without replacement and getting both false negative results is approximately 0.0036 or 0.36%.
To address the second part of the question regarding randomly selecting three subjects and getting all false negative results:
In scenario (a) with replacement, the probability is simply the product of the individual probabilities:
P(All False Negative (with replacement)) = P(False Negative (with replacement)) * P(False Negative (with replacement)) * P(False Negative (with replacement))
P(All False Negative (with replacement)) = 0.006 * 0.006 * 0.006
P(All False Negative (with replacement)) = 0.000000216
So the probability of randomly selecting three subjects with replacement and getting all false negative results is approximately 0.0000216 or 0.00216%.
In scenario (b) without replacement, the calculation follows the same logic as before:
P(All False Negative (without replacement)) = P(False Negative (without replacement)) * P(False Negative (without replacement, second subject)) * P(False Negative (without replacement, third subject))
P(All False Negative (without replacement)) = 0.006 * 6 / 999 * 5 / 998
P(All False Negative (without replacement)) = 0.000018056
So the probability of randomly selecting three subjects without replacement and getting all false negative results is approximately 0.000018056 or 0.0018056%.
In summary, it is unlikely to randomly select three subjects and get all false negative results, regardless of whether the selections are made with or without replacement. The probabilities are very low, around 0.00216% and 0.0018056% respectively.