Sampling Eye Color Based on a study by Dr. P. Sorita Soni at I...

Question

Sampling Eye Color Based on a study by Dr. P. Sorita Soni at Indiana University, assume that eye colors in the United States are distributed as follows: 40% brown, 35% blue, 12% green, 7% gray, 6% hazel.

d. If two people are randomly selected, what is the probability that at least one of them has brown eyes?

d. If two people are randomly selected, what is the probability that at least one of them has brown eyes?

Accepted Answer

Welcome back, everyone. A study on the distribution of fruit types in a grocery store found the following distribution for apples 50% red, 30% green, 15% yellow, and 5% pink. If two apples are randomly selected, what is the probability that at least one of them is red? A says 0.05, B 0.25, C 0.50, and D 0.75. So let's define the probability that we're interested in. We're interested in the probability of at least one red apple when we choose two apples. So if we define red apples as R, we can say that we're looking for the probability of R being between 1 and 2 inclusive. So R is either 1 or 2, right? And for this problem we're going to use the solution that relies on the method of complements. So the probability of R being equal to 1 or 2 is essentially equal to 1 minus the probability of the opposite event, meaning R being exactly 0, right? So what we want to do is simply identify the probability of 0 red apples when 2 apples are chosen, the probability of R being 0. Can be evaluated using the multiplication rule because we are dealing with independent events. Choosing one type of apple doesn't affect. The outcome of our next selection, right? So we can say that the probability of zero red apples is going to be equal to the probability of not choosing a red apple. Multiplied by the probability of again not choosing a red apple, so the probability of our bar once again here we're using compliments and we can just say that this is the probability of the complement of our squared. Now let's identify the probability of not choosing a red apple. We know that we have 50% of red apples, which means that the remaining probabilities add up to 50% as well. We can check that 30 + 15 + 5. This gives us 50%. Which is the probability of not choosing a red apple. And we can say that the probability of our bar is simply 0.50. So now we can identify. The probability of choosing one or two red apples if two apples are randomly selected. Going back to our initial equation, the probability of R being between 1 and 2 inclusive is 1. Minus the probability of zero red apples which is expressed as The probability of our bar squared. And now we can substitute the given value. We get 1 minus 0.50 squared. Which is equal to 1 minus 0.25, and that's 0.75, which corresponds to the answer to the. Thank you for watching.

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Key Concepts

Probability

Complement Rule

Independent Events

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