Based on a research survey, when 1020 adults were asked about hand hygiene, 44% said that they wash their hands after using pu
blic transportation. Consider the probability that among 30 different adults randomly selected from the 1020 who were surveyed, there are at least 10 who wash their hands after using public transportation. Given that these subjects were selected without replacement, are the 30 selections independent? Can they be treated as being independent? Can the probability be found using the binomial probability formula?
Considering that the subjects are chosen without replacement, they are not independent, and the probability cannot be found using the binomial distribution.
The binomial distribution and the hypergeometric distribution are quite similar, as:
They find the probability of exactly x successes on n repeated trials.
For each trial, there are only two possible outcomes.
The difference is that the binomial distribution is for independent trials, that is, in each trial, the probability of success is the same, while the hypergeometric distribution is for dependent trials.
If the sample is without replacement, the trials are not independent, thus the hypergeometric distribution is used, not the binomial.
