Gender of Children A couple has 4 children. Find each probability. a. All girls b. Exactly two g

Gender of Children A couple has 4 children. Find each...

Key Concepts

Complement Rule

The complement rule is a principle in probability that states the probability of an event occurring is equal to one minus the probability of it not occurring. This rule is especially useful for 'at least' scenarios, where it may be simpler to calculate the probability of the opposite (none) and subtract it from one to find the desired probability.

Binomial Distribution

The binomial distribution is a probability distribution used to determine the likelihood of obtaining a fixed number of successes in a given number of independent Bernoulli trials (where each trial has two outcomes). It is particularly relevant for problems that involve counting specific outcomes, such as finding the probability of having exactly a certain number of girls or boys.

Independent Events

Independent events are those whose outcomes do not affect one another. When events are independent, the probability of a combined event is the product of the probabilities of the individual events. This concept is used when determining the likelihood of sequences of outcomes, like the gender of each child in a family.

Sample Space

The sample space is the collection of all possible outcomes of a random experiment. In the context of probability problems, the sample space serves as the foundation for determining probabilities by establishing the total number of equally likely outcomes when an experiment (such as observing the gender of children) is performed.

Probability Theory

Probability theory provides the mathematical framework for analyzing random events and quantifying the likelihood of various outcomes. It involves concepts such as the assignment of probabilities to events, understanding the total possible outcomes (sample space), and applying rules like the complement rule to compute the chances of different scenarios.

Transcript

00:01 All right.

00:01 So the question we're given is that children, a couple has four children, genuine children, a couple has four children.

00:10 Find each probability.

00:13 So the first question is saying, what's the probability that they have all girls when they have four children? well, for each of the children that they have, there is a one and 2 chance of having a girl.

00:34 So that means to find the probability of all four of their children being girls.

00:40 We multiply these probabilities together and along the top one times 1 times one times 1 is one, two times 2 times two times 2 gets us 16.

00:56 So the probability that they have exactly four girls is 1/16.

01:06 And this is because when we are looking at combined probability events, we multiply them together.

01:13 Okay, this would be the same probability of flipping a heads four times in a row, rolling an odd number on a die four times in a row.

01:22 Anything with a one and 2 probability.

01:24 This is how we find the outcome of that happening four times.

01:29 That's a good question.

01:30 Be.

01:32 Question b is asking us finding the probability of getting exactly two girls and exactly two boys.

01:39 Now we could try and draw a probability tree here, but problem is is that it would get quite messy with there being four sets of branches.

01:47 Instead, let's list the outcomes where we get exactly two girls in exactly two boys.

01:54 Now, before we do that, how many total outcomes are there? well, there are four children, so there's four events and for each of them there are two outcomes.

02:10 So to find the total amount of outcomes.

02:13 There are we would do two times two times two times two.

02:17 And that makes 16 outcomes.

02:22 Now, how many of those 16 outcomes result in 2, 2 boys and two girls? well we could have boy boy girl girl, what kind of? boy girl? wait girl, we could a boy girl, girl, boy you could have girl girl boy boy girl boy girl boy or girl.

02:51 Boy boy girl.

02:55 These are all the outcomes that we can find.

02:58 The result in two boys and two girls.

03:02 So that means the probability of them having exactly two girls.

03:06 Exactly two boys would be six over 16.

03:15 Okay, what's the next question? next question is asking us, what's that? probably they have at least one child who is a girl...

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