Question

Recall that for a random variable to be a binomial random variable, you must have an experiment which meets the following three criteria:
1: There are exactly two outcomes for each trial.
2: There are a fixed number (n) of trials.
3: The trials are independent, and there is a fixed probability of success (p) and failure (q) for each trial.

For each of the two situations described below, please indicate if the variable X (as defined in each situation) can be considered a binomial random variable. If you think that X is a binomial variable, please explain how the situation specifically meets each of the three criteria, and identify the values of n and p. If you think X cannot be considered a binomial variable, please indicate which of the three criteria is/are not met (indicate all that apply), and provide a brief explanation for your choice(s). Hint: X can be considered a binomial random variable in only one of the two situations below, but I am not telling you which one, obviously.

Situation 1: A fair coin is tossed over and over again. Let X = the number of tosses until the third TAILS appears.

Situation 2: A box contains 10 marbles: 4 are red, 3 are white, and 3 are blue. A marble is randomly selected, returned to the box, then another marble is randomly selected. Let X = the number of red marbles selected in the two consecutive trials.

Answer 1

Situation 1:

A fair coin is tossed over and over again. the number of tosses until the third TAILS appears.

in each trial there are exactly two outcomes Head or tail.

The trials are independent and there is a fixed probability(p) of getting head and a fixed probability (q) of getting head in each trial.

As there can be any number of trials until the third tails appears. So there are no fixed number of trials.

Hence this experiment does not meet the criteria that there are a fixed number of trials.

so cannot be considered a binomial random variable.

Situation 2:

A box contains 10 marbles: 4 are red, 3 are white, and 3 are blue. A marble is randomly selected, returned to the box, then another marble is randomly selected. Let X = the number of red marbles selected in the two consecutive trials.

Here we can assume that there are exactly two outcomes, (i) RED marble is selected and (ii) RED marble is not selected.

As aa marble is randomly selected, returned to the box, then another marble is randomly selected. So there is equal probability of the selection of the red marble in each trial, i.e. in each trial and in each trial.

the number of red marbles selected in the two consecutive trials.

So there is a fixed number of trials. n=2

As this experiment meets all the criterias of a binomial experiment.

can be considered a binomial random variable. And for this experiment,