So, geneticists use probability in math all the time because they want to be able to determine, you know, what's the chance, or what's the probability that an offspring will have this phenotype, or have this allele, or have this mutation? Any of these things. And so, we have to be able to understand probability laws in order to understand those types of questions. You know, what's the chance that this will happen? So there are 3 laws I want to talk about. The first is the Product Law, and the Product Law is used when you have 2 independent events that are occurring together. So, the example that I'm going to use is coin tosses because that's always used in probability. But if you toss a penny and a nickel at the same time, what's the probability of both being heads?
So, if you toss a penny and a nickel, each has one-half chance of being heads. Correct? Right? The penny has a one-half chance of being heads and or tails. The nickel has a one-half chance of being heads or tails. Each is independent of each other. Whatever the penny lands on has no effect on what the nickel will be, and they're occurring at the same time. They're being tossed together. So what you do is you multiply them. So you take 1/2, multiply it by 1/2, and that equals 1/4 or 25%. So that's when you use the Product Law.
The second one is the Sum Law, and this is, of course, adding because it's the sum. It occurs in independent events occurring together, but you use it when the events can occur in more than one way. So, the example of this is the same example, tossing a penny and tossing a nickel. But what you want is you want one being heads and the other being tails. So there are 2 possible ways that could happen. Right? The penny could be tails and the nickel could be heads, or the penny could be heads and the nickel could be tails. Now, each one of these has a 1/4 chance of happening. How do I know that? Because you have these two options, they could both be heads or they could both be tails, and there's no other combination of these things happening. Right? And, these would both be 1/4 as well. So, in the Sum Law, to understand the probability of one being heads and the other being tails, you add them. So you take 1/4 plus 1/4 equals what? 1/2 or 50%, and that's the Sum Law.
Now, the third one we're going to spend a little more time talking about because there are really two ways to go about doing this, and this is the Binomial Theorem. And this is when there are alternative ways to achieve some type of combination of events. So, the example that I'm going to use is, what is the probability that a family with 4 children, 2 will be male and 2 will be female? There are 4 children, 2 male, 2 female. What's the probability of that happening?
So the first option is using this formula here, \(a + b\) raised to the \(n\). So what do all these stand for? Well, \(a\) is the male probability, which is 1/2. We know this because there are 2 out of 4 males. And \(b\) is the female probability, which is also 1/2 because it's 2 out of 4. So we can put 1/2 + 1/2, but now we need to know what \(n\) is. So \(n\) is actually determined by this. So there are 4 children, so \(n\) is the number of possible outcomes. So here we have 4. Now, you can do this 2 ways. Now, if you are good with math, you can just write it out and do the math and get the formula, which you would input \(a\) and \(b\) for and then get the probability. Or, if you don't feel like doing this, you can use this cheat table. So what this is, is you know that your \(n\) is 4. So here we have 0, \(n\) equals 0, \(n\) equals 1, \(n\) equals 2, \(n\) equals 3, \(n\) equals 4, and \(n\) equals 5. So we know that our \(n\) equals 4, so this is the number that we're interested in.
So what do you do with this? Well, you write out your formula. It's the same formula that you would write out this way, but instead, you're given the coefficients. So the first one is 4. So that is the probability that all 4 of the children will be males. There's a second choice. Right? There's a chance that 3 would be males, but one would be a female, just 4. The third one is 6, and that would be 2 would be males and 2 would be females. The other choice, use this one. So you see we're just going across 1, 4, 6, 4, and this will be 1 at the end. But the next probability is that 1 is male, and 3 are females. And then finally, the last is that all 4 are females, And these are all the options for the children. Now, if I were to do this, fill this in with 1/2 for each one of these, this would equal 1. But we're not interested in what all of this equals. What we are interested in is, what is the probability that a family of 4 children has 2 male and 2 female? So, which one of these represents 2 males and 2 females?
Right. It's this one right here. Because you have 2 males and 2 females. So what you do is you take this, and that is the probability of having 2 males and 2 females, and you just solve for it. So we know that it's 1/2 for \(a\), each \(a\) and \(b\), and so the probability here is 3 eighths. So that is one option of solving this problem. The second option is using a different formula, and that's this formula. So again, you have to solve for \(n\). \(N\) is the total number of events, which we know, right, which we talked about before. So, there's a formula here to calculate it, which is \(s + t\), \(s\) meaning the number of times \(a\) occurs, which in this case is 2 because there are 2 males, and \(t\) is the number of times \(b\) occurs, which is 2 because there are 2 females. So \(n\) equals 4, same thing as above. And so then you use this problem. So \(n\) is 4, \(s\) is 2, \(t\) is 2, and then you have \(a\) and \(b\), which are both 1/2, to the \(s\) and \(t\), which are 2 2. So if you were to solve this problem with your calculator, you would also get \(p\) equals 3 eighths. So it doesn't matter which way you do it. I tend to like this one because I think it requires less math and fewer formulas because, you know, all the coefficients are given to you, and as long as each one of these adds up to 4, then you're good to go. But some people don't prefer that way; some people prefer, you know, just using this formula here to calculate it. But either way, it'll get you that probability. And remember, this is the type of probability when there's some type of combination of events happening, and there are all these different alternative ways that it could happen. So in this case, there are 4 children, and there's, you know, this way they all 4 could be males, 3 could be males, 2 could be males, 1 could be males, or all 4 could be females. A bunch of different alternative ways and combinations of events that could happen in the production of these children. But what you are interested in is that 4, 2 are male and 2 are female. So again, it's the binomial theorem is a little confusing. You may have to watch it again to fully understand. But it is something that you'll likely have at least one question on, on a test setting. So make sure you understand it. So with that, let's now move on.
