Biology: The Multiplicative Law: Some Extensions

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You know the multiplicative law can be used in a lot of different cases. Again, if you understand it, you can use it anywhere you want. You're probably saying, "It's just too much trouble to learn this thing. I just want to do Punnett squares." That's okay, but you have to understand that some genetics get very complex and there's going to be situations when we're doing di- and tri-hybrid crosses in the field of genetics where holy mackerel, how many possible boxes am I going to have to have in my Punnett square. Let me show you some examples. First of all, you can use the multiplicative law to predict merely gametes. Let me give you an example. Let's go to like a di-hybrid. So, here's just a di-hybrid, make up some letters and we could do a problem like this. We could do something like, what are the odds of getting, from this, a SY sperm?
So, this is not a cross. You don't always have to use these in crosses. Now, on the other hand--as you'll see in a minute, we can combine things. But one step at a time. Well, to get a SY sperm, here's the thing. If you take a look here, what are the odds of getting a S out of these chromosomes? Now remember what these represent. They represent chromosomes. This chromosome has the S, this chromosome has the s. This chromosome has the Y, this chromosome has the y. So, in this case of independent assortment, this is very simple. To get a S from this homogulouos pair is one out of two. To get a Y from the homogulouos pair is one out of two. So, therefore, the odds of getting these two thing--and remember what the multiplicative law says, the probability of two independent events both occurring simultaneously. So, what are the odds of getting a S and a Y, one out of two and one out of two. One out of two times one out of two, one out of four.
So, we know that one-fourth of this person's gametes will be SY. Now you can extend that and you can extent that to crosses. Let's do a simple--there's another law that I'm going to tell you about later, but let's just do a simple cross. Well, maybe not such a simple cross. Let's do one--and you know, my student and I remember myself in my first genetics course, just hating these di-hybrid crosses because to the number of boxes I had to draw in Punnett squares and probability solved this for me. So, here's the thing. What are the odds of getting from this combination a, let's just say, a ss, yy offspring?
Now, you know the answer because we've already done this with a Punnett square. We know that it's supposed to be, don't tell anyone I told you this, one out of 16. So, we know it's suppose to be one out of 16, but let's see how we would arrive at that, using the multiplicative law. Well, what are the odds of getting a s? Remember you have to get one of these from each parent. So, what are the odds of getting a s out of dad? One out of two. What are the odds of getting a y out of dad? One out of two. So, what are the odds of getting a sy gamete out of dad? One out of four.
Let's go to mom. What are the odd of getting a s from mom? One out of two. What are the odds of getting a y from mom? One out of two. One out of two time one out of two, one out of four. Now, let's think though. What's the answer? Okay, and here's the thing. The point is, we're asking about two events that are independent of each other. Now we're saying, "What are the odds of getting a sy gamete out of dad?" Which is an event that is independent of getting a sy gamete out of mom. So, we are simply going to use the multiplicative law. One out of four times one out of four equals one out of 16.
Now you could approach that mentally a very different way. Not a very different way, the same way and you could simply say, "Oh, I know." One out of two getting a s from dad, one out of two getting a s from mom. One out of two getting a y from dad, one out of two getting a y from mom. Just approach it differently and then just go right across, two times two time two time two. Sixteen. You see? So, it's so convenient.
I'm going to give you a problem to do and we're going to make it harder still. You guys probably have picked up that I do the easy ones and I give you the hard ones. You learn that in college. I'm going to give you a tough one. I'm going to throw three genes in now. Ready? We're going to do a tri--a three gene cross. We won't make it too nasty. What are we going to do. I'm going to cross--yeah, I'm going to make it a tri-hybrid. All right, here we go. Aa, Bb, Cc. That's that. We're going to cross them with Aa, Bb, Cc. Here's my question, what are the odds of getting an offspring that is aa, bb, CC? What are the odds of getting that offspring? You work on that. You figure out the right answer and then you get back to me.
So, you think you've got an answer, huh? You've heard that before. Let's take a look. What we're looking at is what are the odds of getting a aa. Well, let's approach it this way. To get an a from dad is a one out of two chance, To get an a from mom, one out of two chance. To get a b from dad, one out of two chance. To get a b from mom, one out of two chance. Did this CC thing throw you off? Shouldn't have. What's the odds of getting a C from dad? One out of two. What's the odds of getting a C from mom? One out of two. Two times two times two times two times two times two, one out of sixty-four. Do you want to draw a Punnett square with sixty-four squares? I don't think so.
So, the point is that using this probability now we can really--look how quickly I did that problem. Now there's going to be certain situation, like suppose I had made mom CC. What would be the odds of getting a C gamete from mom? Not one out of two, one out of one. So, we would just consider that a one. So, that would have changed the odds quite a bit, to one out of thirty-two. You've got to work and you've got to think and that's the great thing about this. There's extensions of this. Let me show you one very cool situation.
Let's deal with the trait of sickle cell anemia. We're going to talk a lot about sickle cell anemia in this course because sickle cell anemia is a gene that's very well documented and very widely spread. It's a disease that's quite common in African Americans and indeed in a lot of equatorial Africa, in Africa in general. Sickle cell anemia causes this sickling of your red blood cells, because of a defective gene product which is the protein. But that's not what I want to talk about. I want to talk about the probability here. Sickle cell anemia runs in one out of ten African Americans carry it. In other words, one out of ten African Americans are heterozygous for it. They're heterozygous for it and it's a recessive gene.
I'm going to come back to that later, but for now, it's a recessive gene, at least the way it expresses in the disease form. So, we're going to say, S equals normal and s equals sickle cell. I'm going to ask you a question. Because, again, if you think about probability, you can use this anywhere. Watch this. What are the odds of two African American people randomly chosen from the population, bumping into each other, falling in love, getting married and having a child with sickle cell anemia? Now, they don't know they're genotype. Let's think about that. What are the odds that two randomly chosen African Americans are going to be--so what we're saying--oh, and let me add one thing, neither of them have sickle cell anemia. I don't know that supports, but let's take a look at it.
Let's look at these two randomly chosen people. What are the odds that they're carriers? So, they don't have sickle cell anemia. What are the odds that--I don't want to say that. I want to say, let's just say that they're randomly chosen African Americans. What are the odds that they're carriers? Make it easy, I don't want to be too tough on you. Randomly chosen African Americans, they just bump into each other, know nothing about their genotype, what are the odds that they're carriers? The odds that they're carriers are one out of ten, because one out of ten African Americans are carriers. That being said, what are the odds that this person meets another carrier? One out of ten. So, what are the odds that two randomly chosen African Americans people are going to both be carriers and both meet? One out of one hundred. But if they're carriers that means we need to throw some genetics in here too.
So, let's just say, if they're carriers, therefore Ss crossed with Ss--let's thing about this. What are the odds that their child is going to be ss. One out of two times one out of two. So, what are the odds that this child will be ss? One out of four. So, let's go back to the question. What are the odds that two randomly chosen people will meet and have a child with sickle cell anemia? So, we first calculated what are the odds that they're carriers. Now, we're calculating what are the odds that they're having a child. It is one out of four hundred. Which happens to be the frequency of sickle cell anemia in the population--in the African American population. You can take it still further. What are the odds that they're going to meet on a Tuesday? Times one out of seven because Tuesday is one out of every seven days. So that's one out of 2,800. What are the odds that they're going to meet on a Tuesday in the year 2000? One out of 2,800 times 52. Get it?
So, this whole idea of probability is something that you can use over and over and over again, in many, many ways. Now, you're probably saying to yourself, "Wait a minute. What about situations where there's more than one possible combination?" Ah ha, we're going to get to the additive law later.
Mendelian Genetics and Mutation
Laws of Probability
The Multiplicative Law: Some Extensions Page [2 of 2]