A box contains 10 items, of which 3 are defective and 7 are non-defective. Two items are randomly selected, one at a time, with replacement, and x is the number of defectives in the sample of two. Explain why x is a binomial random variable

https://onlinecourses.science.psu.edu/stat200/node/37

Note in particular:

For a variable to be a binomial random variable, ALL of the following conditions must be met:

* There are a fixed number of trials (a fixed sample size)
* The probability of a success is the same on each trial
* Trials are independent of one another

The variable x is considered a binomial random variable because it satisfies the criteria of a binomial distribution. In order for x to be a binomial random variable, the following conditions must be met:

1. The number of trials, denoted by n, must be fixed. In this case, there are two trials of selecting an item from the box.

2. Each trial must be independent of each other. This means that the outcome of one trial should not affect the outcome of another trial. In our scenario, when an item is selected from the box, it is replaced back into the box before the next item is selected. Therefore, the probability of selecting a defective or non-defective item remains the same for each trial.

3. The probability of success, denoted by p, must be constant for each trial. In this case, the probability of selecting a defective item remains the same for both trials.

4. The variable of interest, in this case, x, represents the number of successes (in this case, the number of defective items) in the n trials.

By fulfilling these four criteria, x becomes a binomial random variable. It allows us to analyze the probability distribution of the number of defectives in a sample of two items selected with replacement from the box.