"In Exercises 1-4, identify the sample space of the probabilit...

Question

"In Exercises 1-4, identify the sample space of the probability experiment and determine the number of outcomes in the event. Draw a tree diagram when appropriate.

1. Experiment: Tossing four coins

Event: Getting three heads"

Accepted Answer

Hi everybody. Welcome back. Our next problem says a family has 4 children. Each child can be a boy, B, or a girl, G. What is the sample space for all possible gender combinations and how many outcomes are there in the event that exactly 1 child is a girl? So recall that that sample space is the set of all possible outcomes. So we're looking at all possible gender combinations, we have two possibilities, boy or girl, with 4 children, we'll have 2 to 4th power or 16. Possible outcomes so we can double check are set against that. So we can start with, for instance, all boys. So no girls. And then we could have the possibility of all girls. And again, you can do this in different orders. What you find is easiest to keep track and not duplicate things. So now, we'll look at possibilities with one girl, which is also the one event we're interested in. Well, that girl could be first, so 1 girl, followed by 3 boys. We could have the girl a 2nd child, so boy, girl, boy, boy or the girl could be 3rd, boy, boy, girl, boy. Or the girl could be last, so 3 boys and then 1 girl. And that makes sense. We'd have 4 possibilities, where exactly 1 is a girl. Because there are 4 positions that girl could be in in our. Overall children. But again, we're not just looking for that. So we've gotten how many outcomes are there where exactly one child is a girl, it's going to be 4. But we also need to list the sample space of all possible gender combinations. We've got our all boys, all girls, one girl. So now, we'll list what when what are the combinations of two girls, two boys. So we could have the girls being first, girl, girl, boy, boy. We have the girls being the two youngest boy, boy, girl, girl. Or they could alternate. We can have girl first, boy, girl, boy. Then with a girl first, we could have girl, boy, boy, girl. With a boy first and a girl second, it could go boy, girl, girl, boy. Or with a girl second you could have boy girl. Boy girl So, let's just keep track of how many outcomes we're up to now, out of our 16, 123456789, 1011, 12, just 4 more. So we've looked at the possibility of no girls, all girls. One girl, 2 girls. We just need the possibility for 3 girls. So we'll start with 3 girls first and then a boy. Then that boy could be in the 3rd position, so girl, girl, boy, girl, or the boy could be in 2nd position. Girl boy, girl, girl. And last of all, the boy could be the oldest, followed by 3 girls, and that gave us our remaining 4 possibilities. So we have our set of 16 total possible outcomes, our sample space. Again, it could be any different order, however, it's easiest for you to make sure you've listed all the outcomes without duplicating anything. So, all our gender combinations. And then we saw that there are 4 outcomes. So there's our set of all possible. There are 4 outcomes in which exactly 1 child is a girl. So, not at least 1 child, but just 1, no more. Hope to see you in the next video.

"In Exercises 1-4, identify the sample space of the probability experiment and determine the number of outcomes in the event. Draw a tree diagram when appropriate.
1. Experiment: Tossing four coins
Event: Getting three heads"

Key Concepts

Sample Space

Event

Tree Diagram

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