A family with 5 children is selected at random. What is the probability that the family has at l

A family with 5 children is selected at random. What is the...

Key Concepts

Binomial Distribution

This concept involves the probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. It is used when the outcome of each trial (such as determining if a child is a boy or a girl) is binary and independent of the others.

Combinatorics and Combinations

Combinatorics deals with counting and arranging discrete objects, and combinations in particular are used to determine how many ways a certain number of successes can be selected from a larger set without considering order. This is crucial when computing probabilities in situations with multiple independent events.

Cumulative Probability and 'At Least' Concept

Cumulative probability refers to the sum of probabilities for a series of events that meet a certain condition, such as 'at least' a specified number of successes. In problems asking for at least a certain count of outcomes, it requires summing the probabilities of all individual events that satisfy the condition.

Transcript

00:01 For the purposes of this problem, we are going to assume that there is an equal chance of having a boy versus a girl for a family.

00:11 Obviously, it's not exactly even 50 -50 shot between a boy or a girl, but for the purposes of this, that's what we're going to go with.

00:21 Besides that, this is a binomial distribution question.

00:26 So we need to find our number of trials, our number of successes, our probability.

00:31 Of success and our probability of failure and then we can solve this out.

00:35 Well, we are told that they have five children.

00:37 Okay, so that is telling us that n is five, right? we've got five people to pick from that sort of thing.

00:46 Our probability of success, p, like i just said at the beginning of this, we are just going to say that there is a 0 .5 chance of success, and we are going to say that there is a 0 .5 chance of failure, meaning one half chance of each.

01:00 So we're going to consider it even chance of having a boy or girl.

01:07 R is the other thing that we need, the number of our successes.

01:12 Well, the question we're being asked here is the probability that we have at least three boys.

01:17 Okay, so that means we don't get to do this just once and be done because it's saying three boys is the lowest that we're willing to accept success -wise.

01:26 So three boys would be a success, meaning we want to find the probability of having three boys.

01:34 But four boys would also be a success because that is not less than three right that's more than three that also means that five boys would be a success now we don't have to go any more than that because remember we only have five kids five children in this family anyways right but we have to find the probability of all three of these and then add them up to get our final answer for this problem okay so what we're going to do is we're just going to be building three separate binomial distributions going off what we've got, what we've been given from our notes up here.

02:13 N is five, total number of children, total number of trials, choose how many successes do we want here? we want three successes.

02:24 Then we have to multiply that by the probability of success, so the probability of getting a boy, which would be one -half point five.

02:34 And that will be taken to the third power because that's how many boys we want.

02:40 But we also have to multiply in the probability of failure, which is also one -half, and if there are five children, and only three of them end up being boys, that would mean two of them are girls, which for the purposes of this problem, would be considered a failure.

02:56 So then we just need to solve all of that out.

02:58 Again, you can either plug that straight into a calculator, or remember if you have a calculator that's got the binomial pdf, on it, then you can also just plug this stuff in.

03:07 Either way, you should get a probability of 0 .3125...

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