Ravi sells real estate. Based on previous data, he knows that $5 \%$ of home tours result in a sale. Assume that the results of these tours are independent from each other.

Which of the following choices are binomial random variables?
Choose all answers that apply:
A Take a random sample of 3 tours and let $K=$ the number of tours that result in a sale.

B Take a random sample of 3 tours and let $M=$ the amount of sales (in dollars) generated by the tours.

C Take a random sample of 30 tours and let $L=$ the number of tours that result in a sale.

Question

Ravi sells real estate. Based on previous data, he knows that $5 \%$ of home tours result in a sale. Assume that the results of these tours are independent from each other.

Which of the following choices are binomial random variables?
Choose all answers that apply:
A Take a random sample of 3 tours and let $K=$ the number of tours that result in a sale.

B Take a random sample of 3 tours and let $M=$ the amount of sales (in dollars) generated by the tours.

C Take a random sample of 30 tours and let $L=$ the number of tours that result in a sale.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the characteristics of a binomial random variable. ‖ ‖ step 2 ⋮ A binomial random variable must have a fixed number of trials, two possible outcomes (success or failure), constant probability of success on each trial, and independence between trials. ‖ ‖ step 3 ⋮ Evaluate option A: A random sample of 3 tours with $K=$ the number of tours that result in a sale. This scenario has a fixed number of trials (3), two outcomes (sale or no sale), constant probability of success (5%), and independence is assumed. ‖ ‖ step 4 ⋮ Evaluate option B: A random sample of 3 tours with $M=$ the amount of sales (in dollars). The amount in dollars is not a count of successes or failures, so it does not fit the criteria for a binomial random variable. ‖ ‖ step 5 ⋮ Evaluate option C: A random sample of 30 tours with $L=$ the number of tours that result in a sale. This scenario also has a fixed number of trials (30), two outcomes (sale or no sale), constant probability of success (5%), and independence is assumed. ‖ ∻1 Answer∻ ⚹ Options A and C are binomial random variables. ⚹ ∻Key Concept∻ ⚹Binomial Random Variable⚹ ∻Explanation∻ ⚹A binomial random variable must satisfy four conditions: a fixed number of trials, two possible outcomes for each trial, a constant probability of success, and independence between trials. Options A and C meet these criteria, while option B does not because it involves a continuous outcome (amount in dollars) rather than a count of successes or failures.⚹◊What is the formula for calculating the probability of success in a binomial distribution?⍭ Generate me a similar question◊

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