In a family of seven children, what is the probability of...

Key Concepts

Symmetry in Probability

When the probabilities for the two outcomes are equal, as in the case of a 50/50 chance for boy or girl, the scenarios display symmetry. This means that the probability of getting a certain number of boys is the same as the probability of getting the same number of girls, which simplifies the analysis and calculation of probabilities in such experiments.

Independent Trials

The concept of independent trials means that the outcome of one trial does not influence the outcome of any other. This is critical in probability calculations for events like the gender of children, where each child’s gender is independent of the previous ones, ensuring that the binomial model applies.

Probability Mass Function (PMF) of the Binomial Distribution

The PMF provides the formula to calculate the probability of observing exactly k successes in n independent trials. It is given by the expression P(X=k) = nCk * p^k * (1-p)^(n-k), where p is the probability of success on each trial. This function is foundational in calculating the probabilities of various outcomes in problems involving binary events.

Combination Formula (nCk)

The combination formula is used to determine the number of ways to select k successes out of n independent trials, regardless of the order in which the successes occur. It is expressed as nCk, and it plays a crucial role in calculating the probabilities associated with specific outcomes in a binomial experiment.

Binomial Distribution

This concept refers to the probability distribution that arises when there are a fixed number of independent trials (such as births), each with two possible outcomes (for example, boy or girl), and each outcome has the same probability. In such cases, the number of successes (or one of the outcomes) can be modeled using the binomial distribution.

Transcript

00:01 For this problem, we're being asked to figure out the probability of this family of seven having different variations of boys versus girls among their kids.

00:13 And what we need to know in order to solve each of these different parts to this problem is number one, the probability of getting either a boy or a girl, which we can assume is roughly 50 -50 for the purposes of this question.

00:27 The second thing we have to figure out is how much.

00:30 Different ways can we get, let's say, let's look at option e, four girls and three boys.

00:36 How many different variations of that order can we get among seven kids? and we have to figure out the probability of each of those instances.

00:46 And then the final thing is the probability of all of them combined.

00:50 And so let's take a look at an example of that.

00:53 So for part a, excuse me, we have each of these seven kids designated by these little circles and they were all boys.

01:03 And each of them had a 50 % chance or a probability of one half of being a boy.

01:11 And so in order to find the probability that all of them would have this probability, this final outcome of being a boy, we would multiply each instance with every other instance.

01:33 And if we wanted to simplify this a little bit, that would look like one half to the seventh, because we're multiplying one half by itself seven different times for each of the seven kits.

01:47 And once we work that out, you can plug this into a calculator if you'd like, or just work it out by hand, we would get one over 128.

02:00 So there was a one out of a 128 chance that all seven kids are boys.

02:08 That seems pretty straightforward in terms of solving, right? okay.

02:13 Let's move on to option b, or part b, excuse me, which is the same thing almost, except in this case we're asked the probability that all children are the same sex.

02:27 So not only do we have the probability of them being all boys, which we solved for up here, as we saw that was one, over 128.

02:37 We also have another option for how this could happen.

02:41 So the desired outcome is all children of the same sex.

02:45 There are two different ways this could happen, given the confines of this question.

02:50 The first would be if all of them were boys, but the second would be if all of them were girls.

02:55 And so we have to calculate the probability of each one of these different occurrences and add them together to find the total probability of that event occurring.

03:06 So we have already have our probability for them all being boys.

03:13 And if you have noticed, our probability for them all being girls is the same thing.

03:20 It's one half to the seven or one over 128.

03:24 And so all we wanted to know was the probability of either of these events occurring, we would simply add them up.

03:36 So we have 1 over 128 plus 1 over 128.

03:42 Add that together, we get 2 over 128.

03:44 And if you wanted to simplify that down a little bit further, it would be 1 over 64.

03:50 So now we have greatly increased the likelihood of our probability just by changing the desired outcome slightly.

04:04 Now what about option or part c, excuse me, keep saying option.

04:09 It's because we're dealing with probability.

04:11 We have a lot of different options here.

04:13 For part c, we have a description of an event that might seem simple, but as you can tell from this diagram, isn't so easy to figure out because all it says is the probability that there will be six girls and one boy among these children.

04:36 But that doesn't tell us what order they're going in.

04:40 So that means that the boy could be the first child.

04:44 He could be the second or the third, the fourth, et cetera.

04:47 We don't know, so we have to account for all of these possible variations or permutations, as that's what statisticians call this, in order to find out the probability of there just being one boy in this family of seven kids.

05:08 And so in order to do that, we have to know the probability of each of these things happening.

05:17 And so, let's see, the probability of this one child being a boy is one half.

05:24 Same thing for this child being a girl.

05:27 And we could repeat this process throughout all of these different children, but i can tell you that at each step, we're going to have the same equation to solve of one -half to the seventh, or one over 128.

05:46 And so each of these different permutations has a probability of one over 128 of happen.

05:59 And if this is happening 78 times, or seven times, getting a little ahead of myself here, this is happening seven times, then we are adding 1 over 128 to itself seven times, which would equal 7 over 128, which i do not believe simplifies any further.

06:20 So this is a good example of how we can go from what seems to be a simple event, six girls, one boy, to having to draw out multiple permutations using our multiplication rule in order to find out the probability of each different permutation happening, and then adding them all together to find out the probability of the event occurring, the event being having six girls and one boy.

06:50 So you might be thinking, all right, well, i guess i have to draw out all of these permutations for every single thing.

06:56 So let's go to the next one.

06:58 Four boys and three girls.

07:01 You might be thinking, well, with one boy, we already had seven different lines.

07:07 With having four boys, and who knows how many different variations of that happening, that might be way too many permutations to draw out by hand, and you're probably thinking there has to be an easier way than doing this...

Please give Ace some feedback