Which of the following choices can be considered binomial random variables?

Choose all answers that apply:

A A container holds 20 pens, half of which are red. Select an SRS of 5 pens and let $Q=$ the number of red pens obtained.

B A container holds 1000 pens, half of which are red. Select an SRS of 5 pens and let $S=$ the number of red pens obtained.
c A container holds 100 pens, half of which are red. Select an SRS of 5 pens and let $R=$ the number of red pens obtained.

Question

Which of the following choices can be considered binomial random variables?

Choose all answers that apply:

A A container holds 20 pens, half of which are red. Select an SRS of 5 pens and let $Q=$ the number of red pens obtained.

B A container holds 1000 pens, half of which are red. Select an SRS of 5 pens and let $S=$ the number of red pens obtained.
c A container holds 100 pens, half of which are red. Select an SRS of 5 pens and let $R=$ the number of red pens obtained.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the criteria for a binomial random variable. A binomial random variable must satisfy the following conditions: 1. There are a fixed number of trials, $n$. 2. Each trial has only two possible outcomes: success or failure. 3. The probability of success, $p$, is the same for each trial. 4. The trials are independent. ‖ ‖ step 2 ⋮ Evaluate each scenario to see if it meets the criteria for a binomial random variable: - **Choice A**: A container holds 20 pens, half of which are red. Select an SRS of 5 pens and let $Q=$ the number of red pens obtained. - Fixed number of trials: Yes, $n=5$. - Two possible outcomes: Yes, red or not red. - Probability of success: No, the probability changes as pens are selected without replacement. - Independence: No, the trials are not independent due to the changing probability. - **Choice B**: A container holds 1000 pens, half of which are red. Select an SRS of 5 pens and let $S=$ the number of red pens obtained. - Fixed number of trials: Yes, $n=5$. - Two possible outcomes: Yes, red or not red. - Probability of success: Approximately yes, since the population is large, the probability of selecting a red pen remains nearly constant. - Independence: Approximately yes, for a large population, the trials can be considered nearly independent. - **Choice C**: A container holds 100 pens, half of which are red. Select an SRS of 5 pens and let $R=$ the number of red pens obtained. - Fixed number of trials: Yes, $n=5$. - Two possible outcomes: Yes, red or not red. - Probability of success: No, the probability changes as pens are selected without replacement. - Independence: No, the trials are not independent due to the changing probability. ‖ ∻Answer∻ ***B*** ∻Key Concept∻ ⚹Binomial Random Variable Criteria⚹ ∻Explanation∻ ⚹A binomial random variable requires a fixed number of trials, two possible outcomes, constant probability of success, and independent trials.⚹◊What is the formula for calculating the probability of obtaining a specific number of red pens in a binomial distribution scenario?⍭ Generate me a similar question◊

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