In Exercises 29–32, use the given sample space or construct th...

Question

In Exercises 29–32, use the given sample space or construct the required sample space to find the indicated probability.

Four Children Exercise 29 lists the sample space for a couple having three children. After identifying the sample space for a couple having four children, find the probability of getting three girls and one boy (in any order).

Accepted Answer

Welcome back, everyone. A person adopts 4 pets, and each pet is either a cat C or a dog D, with both being equally likely. List all possible outcomes in the sample space. What is the probability of getting 3 cats and 1 dog in any order. So first of all, let's understand that our N value is 2. We have two possible elements C and D, and our K is equal to 4. We're trying to form a set of 4 animals, right? So the total number of outcomes is going to be N to the power of K, which is 2 raises to the power of 4, and that's 16. So we now have an idea about our sample size S. It is going to have 16 combinations. Now, how do we form it? Well, we can begin with adopting 4 cats, which gives us 4 Cs. That would be C, C, C, C, right? Now, this is the only possible way because the order doesn't matter. The next possibility consists of 3 cats and 1 dog, so we can do that by having 3 cats followed by 1 dog. And then we're just changing the position of the dog. We can put it at the 1st, the 2nd 1, the 3rd, or the 4th position within our arrangement, which means that the next one is going to be CCDC. See the CC And then D C C C. Well done, so we have 4 of those. Now we have to consider 2 dogs and 2 cats. Let's begin with 2 cats, which gives us CCDD. We can then move one of the Ds to the left, which gives a CD CD. Let's move it to the left again, which gives us D, C, C, D. Then let's move the fourth letter to the left, essentially we're going to move that last D to the left, which gives us the CDC. Then we can move it to the left again, which gives us the DCC, and we can also have two adjacent D letters in the form of CD, DC, right? Well then, so we have 123456 of those. Now let's assume that we adopt 3 dogs, so 1 cat followed by 3 dogs. Then we can have D C D D. And let's keep moving C to the right, which gives us the the CD and DD. D C Finally, we're going to have 4 dogs. The only possibility is to have 4 theses. Now, let's count those. We have 12345678, multiplied by 2, that's 16, so we have our sample space. And now let's identify the probability of an event A. What is the probability of getting 3 cats and 1 dog? So let's recall that it is equal to the number of favorable outcomes divided by the total number of outcomes. Our total is 16, and the number of favorable outcomes can be found from the sample space. We want 3 cats, so C C C D C C D C C D C C. DCCC, and we can see that we have 4 of those, meaning AS 4. Therefore, our probability is 4 divided by 16, which is 0.25. Well done, so let's label our answers, our probability and the sample space for this problem. That will be it. Thank you for watching.

��app

Key Concepts

Sample Space

Probability

Combinatorics

Watch next