Question

You took a SRS ( Simple Random Sample)  of size 40. Suppose that another student presented the exact same data values as you. Calculate the probability of this happening. Should I as the instructor be concerned?

Answer 1

SRS without replacement

Probability of second student getting the exact same data values is

SRS with replacement

Probability of second student getting the exact same data values is

In both the cases if the population size N is large, the probabilities will be very close to zero. So the instructor should be concerned as probability of happening this is too small.

For example if N=50, the probabilities will be smaller than 0.0000000001. ie . In N > 50, the probability will be even smaller

Calculations

Suppose we have population size N and we are taking a simple random sample of size 40.

Case - I Consider SRS without replacement

Suppose the first student got the sample .

Suppose the second student also got the same sample data values.

Then second student got in the sample. This can be selected at any one of the 40 draws and hence has probability of selection 40/N.

Again second student got in the sample. This can be selected at any one of the 40-1=39 draws and hence has probability of selection 39/(N-1).

Proceeding like this

second student got in the sample. This can be selected at any one of the 40-39=1 draws and hence has probability of selection 1/(N-39).

Hence probability of getting this same sample is

This probability

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Case -2 Consider SRS with replacement

Suppose the first student got the sample .

Suppose the second student also got the same sample data values.

let be the ith unit selected in the sample. This unit can be selected in the sample either at first draw, second draw, ..., or nth draw. At any stage, there are always N units in the population in case of SRSWR, so the probability of selection of at any stage is 1/N

Hence probability of getting this same sample is

This probability