Suppose that each child born to a couple is equally likely to be a boy or a girl, independently

step 1 Hello, everyone welcome to the channel. This is tim. Your host i'll be fall, be taking you throu

Suppose that each child born to a couple is equally likely...

Key Concepts

Uniform Probability

This concept refers to scenarios where every possible outcome has the same likelihood of occurring. In this context, since each child is equally likely to be a boy or a girl, the assumption of equal chances simplifies the computation of probabilities across the whole sample space.

Independent Events

Independence means that the outcome of one event does not influence the outcome of another. Here, the sex of each child is determined independently, which allows us to multiply probabilities of individual children when calculating the probability of a specific combination of sexes.

Sample Space

The sample space is the set of all possible outcomes of an experiment. When dealing with multiple independent binary events, like the sex of each child, the total number of outcomes is 2 raised to the power of the number of events (for example, 2^5 for 5 children), providing a framework for calculating probabilities.

Binomial Distribution

This distribution applies to experiments with a fixed number of independent trials with two outcomes per trial (often termed as success and failure). It is used to calculate the probability of obtaining exactly k successes (e.g., boys) in n trials (e.g., children) using combinations, which is particularly useful for parts of the problem like finding the probability of exactly 3 boys.

Combinatorics

Combinatorics involves counting the number of ways certain outcomes can occur, such as choosing which children are boys or girls. Utilizing combinations (often represented as 'n choose k') is key in determining the number of favorable outcomes when the order of selection is not important.

Complement Rule

The complement rule is a probability principle stating that the probability of an event occurring is equal to one minus the probability of it not occurring. This is particularly useful for calculating probabilities such as 'at least one girl' by subtracting the probability of having no girls from one.

Transcript

00:01 Hello, everyone welcome to the channel.

00:03 This is tim.

00:04 Your host i'll be fall, be taking you through this question suppose this comes from problem 25 of your probability text book and we we rear the problem little bit here using our notations just said that it's clear that we're not copying and pasting the problem from the text and we will take a close look at this problem so without further ado.

00:36 Let's begin suppose that we have a boy and a girl that are born.

00:43 There are possible events that are born to a couple and that there are 4 such children that are born.

00:56 So if you're, geneticist and you're trying to figure out what the following probabilities would be, this would be very important down the line for those who would want to know what the probability of family sins might be in the future.

01:18 So if we take a look at the first example, we know that there's going to be a probability of 2 cases occurring in all 4 situations.

01:28 So since the probability space stays the same, it doesn't change independent of 1 another.

01:34 Then we can begin to think about what happens when we combine those events that are independent of 1, another into a space, so, for instance, since there's 2 possible outcomes in every event and they're each independent, they don't depend upon 1, another 10, the boreal every time and then we can multiply these events and see what the probability is base would be.

02:01 So in this situation we can see that there's going to be 1 out of 2 times 1 out of 2 situations.

02:16 It's gonna be a girl times 1 over 2.

02:22 Finally, times 1 or 2 point now: let's see if this is equal to 1.

02:30 Over 16 point you can see once we've written all of them out.

02:35 You can see, there's only 1 situation that all girls will occur.

02:40 This makes sense both mathematically as well as practically.

02:48 We assume that there is a greater probability of having 2 boys and 2 girls than just all girls give theirs for possible siblings.

03:00 So, let's take a look at this and let's calculate or let's just take a look at what this probability might look like and we're gonna look at exactly 2 boys and 2 girls.

03:15 We're not going to take the same first tide above here.

03:17 Take a little too long, we're going to apply this chart and we're going to see that there's only 6 out of 16 possible outcomes that have exactly to boys integrals.

03:35 So i urge you to do something like a chart to confirm the algebra.

03:40 You may be doing at this point of time to confirm for yourself.

03:44 We need to do ay, so just as we've got just as we've gone through the situation with all girls and except our girls, and we got on 1 under 16 point exactly to what integral got 3 out of 8.

04:01 It we're going to did each 1...

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