In Problems 7–16, determine which of the following probability...

Question

In Problems 7–16, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why.

Three cards are selected from a standard 52-card deck without replacement. The number of aces selected is recorded.

Accepted Answer

Welcome back, everyone. In this problem, a classroom has 40 students, 6 of whom are left handed. 4 students are chosen at random without replacement, and the number of left-handed students among the chosen 4 is recorded. Is this a binomial experiment? Select the best answer. A says yes, because there are only two possible outcomes, left-handed or not, for each child. B says no because the trials are not independent. C says yes because the number of trials is fixed at 4, and the D says no because the probability of success changes with each child. Now, to determine whether this is a binomial experiment, we first have to ask ourselves, what do we know about these types of experiments. Well for starters recall that a binomial experiment has a fixed number of trials. Camp We know that it has to have two possible outcomes, thus the name binomial. We know that it has to be independent, OK, or it needs independence. And we also know that there needs to be a constant probability of success. So what we need to do in this scenario is to look at our problem statement and see if it meets this criteria. Now let's start with the trials. Is there a fixed number of trials? Well, in this case we're told that 4 students are chosen at random, OK? So because we choose 4 students, that means in this case the this number of trials is fixed, OK? So it satisfies that criteria. Next, we are told that we're trying to determine if the students are left handed, so either the students are left handed or not. Thus there are two possible outcomes because each trial results in one of two outcomes. Now what about independence? Well, the outcome of one trial should not affect the others. That's what independence means. But in this case, while there are two possible outcomes, the trials are dependent because once a student is selected, the composition of left-handed and right-handed students changes, thus affecting the probability of success and subsequent subsequent trials. So the probability of selecting a left hand student is not constant and thus this is not, this experiment does not guarantee independence. The probability of selecting a left hand student should remain constant, so neither does it guarantee a constant probability of success. So we know then that this is not a binomial experiment. So which one of our answer choices is best suited for our result? Well, in answer twice A, it says yes because there are only two possible outcomes for each child, but we've already seen that this is not the only thing that would make this a binomial experiment. In answer choice B, it says no because the trials are not independent. So far this is the best answer we can test that by reviewing answer choices C and D. No C says yes because the number of trials is fixed at 4, but again this is not the only requirement for it to be a binomial experiment, so C would be incorrect. And finally in B it says no because the probability of success changes with each child and the probability of success does change with each child because the nature of the sampling without replacement. So this is not the, not sorry, a binomial experiment. Therefore, B is the best answer. Thanks a lot for watching everyone. I hope this video helped.

In Problems 7–16, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why.
Three cards are selected from a standard 52-card deck without replacement. The number of aces selected is recorded.

Key Concepts

Binomial Experiment Criteria

Independence of Trials

Sampling Without Replacement

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