Mendel’s Peas Mendel conducted some of his famous experiments ...

Question

Mendel’s Peas Mendel conducted some of his famous experiments with peas that were either smooth yellow plants or wrinkly green plants. If four peas are randomly selected from a batch consisting of four smooth yellow plants and four wrinkly green plants, find the probability that the four selected peas are of the same type.

Accepted Answer

Welcome back, everyone. A group of 8 students consists of 4 who prefer to study in the library and 4 who prefer to study at home. If 4 students are randomly selected from the group, find the probability that all 4 selected students have the same study preference. A says 0.550. B 0.117. C 0.879 and D 0.029. We're looking for the probability. Let's call it an event A. A represents an event that we're choosing for students who have the same study preference, and we have to recall that probability is simply the ratio between two numbers M and N. Where Is the number of favorable outcomes. And N is the total number of outcomes. So let's model this situation. We have 4 students who prefer to study in the library, so we're going to label it as L. We also have 4 students who prefer to study at home. Let's call it 8. The total number of students is 8. And now using this model we can determine M and N values. So the total number of outcomes and. It is going to be found by the total number of ways to select 4 students out of 8. We have to understand that this selection is taking place without replacement and the order does not matter. So we can use the N choose K formula, right? And specifically just not to confuse ourselves with these notations, we're going to change. And to some other letters such as P, right? We can do that. And now we have to recall that and choose K. Can be calculated as unfactorial. Divided by key factorial, which is multiplied by N minus Kfactorial. So now our P. is equal to 8, the total number of students choose 4, right? Because we're choosing 4 out of 8, and this gives us the total number of ways to choose 4 students out of 8 with without replacement, and when the order does not matter. Now what is the number of favorable outcomes um? Well, we have to choose 4 that have the same study preference. So we either choose all 4 who prefer to study in the library or all 4 who prefer to study at home. So M will consist of two different combinations. We either choose 4 out of 4 who prefer. To study in the library, so that would be 4 choose 4 or meaning where basically like adding 4 out of 4 who prefer. To study at home. So again, 4, choose 4, right? And essentially, if N and K are the same, this gives us 4 choose 4 equal to 1, right? Because our N and K are equal to each other. In other words, M is just equal to 2 because we have 1 plus 1. So the probability of A is equal to M, which is 2, divided by P, which is 8, choose 4. And according to the formula that would be 2 divided by N factorial, which is a factorial. Divided by K factorial, which is 4 factorial, which is multiplied by 8 minus 4 factorial, so multiplied by 4 factorial. When we perform this calculation, we end up with 0.029. Meaning the correct answer to this problem corresponds to the answer choice D. Thank you for watching.

Key Concepts

Probability

Combinatorics

Binomial Distribution

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