A population of 300 voters contains 147 Republicans, 135 Democrats, and 18 independents and members of other parties. A simple random sample of 30 voters will be drawn from this population. Consider repeatedly drawing simple random samples of size 30 from the population (taking each simple random sample from the full population of 300 voters). The expected value of the number of simple random samples of size 30 that must be drawn until (and including) the first sample that contains at least one voter who is neither a Democrat nor a Republican is _______ ?
To find the expected value of the number of simple random samples that must be drawn until (and including) the first sample that contains at least one voter who is neither a Democrat nor a Republican, we need to determine the probability of drawing a sample that meets this condition in each round.
Let's first find the probability of drawing a sample that contains at least one voter who is neither a Democrat nor a Republican.
Total number of voters = 300
Number of Republicans = 147
Number of Democrats = 135
Number of independents and others = 18
To find the probability of drawing a sample without any independents or members of other parties, we need to find the probability of selecting only Republicans and Democrats in the sample.
Probability of selecting a Republican = Number of Republicans / Total number of voters = 147 / 300
Probability of selecting a Democrat = Number of Democrats / Total number of voters = 135 / 300
Since the sampling process is done with replacement, the probability of not selecting an independent or member of another party in one draw is given by:
Probability of not selecting an independent or member of another party in one draw = 1 - (Probability of selecting an independent or member of another party in one draw)
Probability of not selecting an independent or member of another party in one draw = 1 - (Probability of selecting a Republican + Probability of selecting a Democrat)
Probability of not selecting an independent or member of another party in one draw = 1 - (147/300 + 135/300)
Now, let's consider the probability of not selecting an independent or member of another party in each sample of size 30.
Since each sample is independently drawn, the probability of not selecting an independent or member of another party in each sample will be the same. Therefore, the probability of not selecting an independent or member of another party in any single sample of size 30 is:
Probability of not selecting an independent or member of another party in a sample of size 30 = (Probability of not selecting an independent or member of another party in one draw) ^ (sample size)
Probability of not selecting an independent or member of another party in a sample of size 30 = [(1 - (147/300 + 135/300)] ^ 30
Finally, the expected value of the number of simple random samples that must be drawn until (and including) the first sample that contains at least one voter who is neither a Democrat nor a Republican can be found by calculating the reciprocal of the probability obtained above.
Expected value = 1 / (Probability of not selecting an independent or member of another party in a sample of size 30)