Question

A die is thrown repeatedly until a six comes up. What is the sample space for the experiment?

Answer 1

Sample space will contain infinite number of sample points. You can have infinite number of non-6 throws before getting 6. Probability of such event will be infinitesimaly low but that will be a valid sample point.

This experiment will produce geometric distribution. X=Geom(1/6)

Generally, geometric distribution tells us probabilty of an event with number of failure before success. In this case, success if getting 6 and failure is any possible outcome but 6.

Probabilty of kk failure before success is given by,
P(X=k)=p(1−p)^k

where, pp is probability of success and (1−p)is probability of failure. In this example, p=1/6and (1−p)=5/6.

Therefore, P(X=k)=(1/6).(5/6)^k

As you can notice, when number of failure increases, probability of that event decreases. Therefore, probability of having many failures is very low but still that event is element of sample space. Hence we can have infinite number of samples in sample space.

Hence, the sample space of this experiment is given by

S = {6, (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (1, 1, 6), (1, 2, 6), … , (1, 5, 6), (2, 1, 6), (2, 2, 6), … , (2, 5, 6), … ,(5, 1, 6), (5, 2, 6), …}