Is this a Random Variable on the sample space?

Question

I am confused about identifying random variables.

For instance, I have question that says "You roll a die until you roll a 6 a total of four times, and you record the resulting sequence. Which of the following are random variables on the sample space? 1. The number of rolls required for the experiment. 2. The sequence resulting from the experiment 3. The time it takes to complete the experiment 4. The number of times a 5 is rolled

SO, the sample space is the sequence of rolls. So for example, one item in the sample space could potentially be (1,3,4,6,6,5,5,6,4,6) while another item could be (5,5,5,3,2,6,6,4,1,2,1,6,5,4,3,6) Basically just an infinite number of sequences. Since RV are supposed to map to the sample space, and the sample space is a sequence, I think that #1 and #3 are not random variables,but that #2 and #4 are random variables. My friend disagrees. Any thoughts? I know that #1 sounds like a random variable but I feel like it is a trick because it doesn't map to our space.

Thanks!

Answer 1

1. The number of rolls required to roll a 6 a total of four times, is indeed a random variable. For example, you may roll a 6 four times in a row the first four rolls. On the other hand you may not even roll the first 6 until the 100th roll. You could indeed calculate the probability of when this will occur.

The probability that it happens after four rolls, the probability that it happens after five, and the probability that it happens after n (arbitrary) rolls.

2. A sequence resulting from one experiment is simply a sequence, not a random variable.

3. This seems to be the same as 1.

4. The number of times a 5 is rolled before you complete the experiment is a random variable. You could compute that probability of rolling only one 5, rolling two fives, etc, before you complete the experiment.