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A family has two children. Determine(a) the event E1 that at least onechild is a boy; (b) the event E2 that at least onechild is a girl. (c)Are E1 and E2 mutuallyex...

Question

A family has two children. Determine(a) the event E1 that at least onechild is a boy; (b) the event E2 that at least onechild is a girl. (c)Are E1 and E2 mutuallyexclusive? These are sets of possible occurrences, so use braces.

A family has two children. Determine (a) the event E1 that at least one child is a boy; (b) the event E2 that at least one child is a girl. (c) Are E1 and E2 mutually exclusive? These are sets of possible occurrences, so use braces.



Answers

A couple has 4 children. Find each probability. a. All girls b. Exactly two girls and two boys c. At least one child who is a girl d. At least one child of each gender

Yeah. Yeah. Hello everyone for this session we're going to take a look at problem, we're presenting with four Children and assuming that either board girl, we want to see the probabilities of each one of those situations. So one easy way to do this would be to write out All possible um 16 combinations. So since 16 is not a large number, you can see that the probability for all growth is uh there's only one, right, there's only one, so B one out of 16. Similarly take a look at this list. If you're looking for exactly two grows and two boys, we can see this only, this only occurs for six of these combinations, so B six out of a total 16 and I would reduce this down to um three over eight. I just showed you that. Okay, so as you can see Um this next one um you can actually apply rule here. So we can either add up all the situations where there's one at least one girl or we can notice that there's only one situation where there are no girls. So in this situation It simply be 16 minus one, which is 15, I would have 15 out of 16 boys and girls. I mean, yes, There's at least one girl. Okay, now, if you take a look at this last one, uh we can apply the same, we can apply the same reasoning. Um We're going to put the same reasoning as a previous problem to this problem as well. So um it says there's a in this province after at least one child of each gender. So if they're all boys off their all girls are going to exclude those. Um And if there's at least one boy and one girl were gonna include those. So one way to write one way to do this would actually be very similar to what we did before, which is um, consider 16 -2. Which is the all girls are all boys. Give us, um, probability of 14 out of 16 or uh huh. Seven. Our turf. Eight. Yeah. I hope you enjoy the session, The 4 to see you next time.

Yeah. So we have, you know, before the situation, we have family. Well, he has, uh, in Children. Children s. Oh, you want to look at the events for the event? Yeah, E There is the event that are the family of as, uh a child's, uh, well, Children off both sexes, both on success on the the over the world. It's not is the world on the other event F eyes that does most most one one. Uh, boy. So I'm going to look at the probability of these events. Yeah. Probability on there now for several choices off the number off child in the family for several options of them, probably of this. And then we want to check. These are I mean on there are independent. So being independent means that the probability off the event okay on the event f to happen simultaneously is equal to the probability off. E times the probability off there. So this is equality here. This is, uh, the property off being independent, being independent events, being independent. Yeah. So let's consider first probably deal. Mm. The probability off I'm compared to Probably pee on death. The project be there for several choices. Well, then so in the case, first first case, you consider what happens if you have to the child. So the possibilities are all that we have the facility is that we have avoided a boy. We have a boy. My bill on the other option is that we have a girl. Why not go? So we're going to count vision because they were not considering without considering the like, who's the older or like this event could be realized in two ways. Like the boy comes first. Something the other girl, um, is born after or communicates Ethical is born first, and then the boy oh is born after. But we're gonna count those us the same event because, no, you don't want to make a distinction on which one is the older. So like, off the permutations off these are all the possibilities. And so the positive the event is that we have both sexes. So I only realized, like this one. So as we can see the probability off e, there's gonna be all day the size of the like. This is the total sample space, which we can call the, so it's gonna be the size of way number The realization off devotee divided by the size of the So. So that is what would be one third. Because there's three possibilities. We have only one realization of that on the probability off F Yeah, that we have, um, most 11 Boy, you have this event on that event, Surat. If he's gonna be too thirds on the probability. But they went down. He on the f happens. Yeah, both sex, both sexes on the at most one child is just this one. So it would be one third on the war. As we can see, it is not equal to bro. Give f times the worldly guilty because that number would be one third times two thirds, which is called toe to over nine, which is which is not one third. So they're not independent in this case, Not in the present. Now, what happens if we consider now family off for, uh, foretells? So the number of possibilities here B is going to be that we have Oh, for boys with vindicated. We have three boys, one girl, or you could have just two boys. Two girls t o B to G or even know one boy and three girls or we could I'll have all all the off the child's our girls. So then, in this case are the probability of B. If you have both sexes will be this event that have been this one. So it's gonna be he's gonna be these three realizations to the three. Off of all this is 1 to 4. Uh, well, bye. Three or five. The probability of f that are most one um, it's a boy would be just these two events should be too some Wait, these three r e and then these two are f So the two for five on the the probability that we have e on at the same time. So both sexes on then. At most one. One child is, uh, which is this one. So that would be, uh, once over or five possibilities. So So this would be one of our fight on the wall. As you can see again, that that is not even apply. These two is not to over five. That's 3/5, because this would give us This is goto six or five, which is not equal to one or five. So this case are they're not independent either. The case off having four Childs not independent on the Now, let's consider the case that we have five kids. See, we have an eagles family off with five kids. It was really These are that they have all off. Five of them are boys, or four of them are poised. And then one girl, we have three boys and two girls or the opposite. Two boys on three girls, three G or we could have a boy and four girls. But then the last second. Is that all? All those. All those shells are just girls. So in this case, we're gonna have six. So 123456 six events, Which is the tall possibilities on, then the probability off key. But we have both sexes. Would be 12 three, four. Right, So we have 4/6 todo to thirds on the probability off the event. Yeah, that is that is that we have at most one. One boy is that we have this one vision on that one. These tools, um, I'm gonna be dead and death, so it will be too over, um, all six. And then the probability that we have e there is you have both sexes on that most one boy is only this realization. This event is that we are both on the most one boy. That means have one boy under there is girls would be just this event, the 1/6 one or six and then even multiply. All this number is ableto well, she would be one third onda. Uh, this number is not equal to probably killed. Yeah, I was the probability off, because this number is equal. Thio two thirds. Thanks. One third, which is equal toe to over nine, which is not ableto 16 chicken. So, uh, neither in this case, they are independent sel not independent better. Yeah, I find it. So, um, this event, the event of Havina Childs of bold sexes on the event that are at most one of them, One of the child is a boy. They're not independent for the case that we have to for the case that we have, uh, toe toe childs. And then for then it is not independent either. In the case of the five Childs

So in a family of five Children, what is the probability that all of them our boys? So if there's an equal chance or any colitis of having either boys or girls? Uh so it means that there's a half chance for having either sex. So the probability is half, we would use permutations, but all boys would be after the power of five, which is one over 52 1/32. Okay then the probability that at least there's one girl, so having one girl means that you can have 12 or three or four or five girls. So it's quite a long process. So we can just use the compliment and say um one minus the probability of no girl, which will give us one minus what is the probability of zero girls? It has already been given as being all boys. It's 1/32 from our a calculation when this would be. So this will be 32/30 to minus one over 32 which is there to one out of 32. So those are the two possible answers for this question.

So we have a sample space, um not gonna copy it down, but it is um has a magnitude of eight. This means that there's eight in the sample size and were asked what is the probability that we have exactly one girl, So exactly one G. And there is three of them out of the eight that do. And this translates to be a 0375% chance, 37.5 chance, otherwise known as 0.375 in terms of probability.


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