Biology: Laws of Probability: The Additive Rule

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11/14/2010

~ Himanshu

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You know. I've given you some problems that I know you've worked through and are starting to seem pretty easy to you, but I can hear some of you saying wait a minute. He's skipping some things. For example, I gave you a problem like this. `Ss' crossed with `Ss', and I gave you the easy one. I said figure out the odds using probability of it having an `ss' offspring. And you said well, that's easy. That's one out of two times, one out of two equals one out of four. So I went on to another one, and I heard someone screaming wait a minute. What about this? `Ss' crossed with `Ss', you didn't ask us, how could you get a `Ss' out of that? How could you get a heterozygote, and the reason I didn't ask you that question is because that's a little bit more of an advanced topic. Why? Well, let's see why. See, in this kind of situation, we could get, to get this baby, we could get a `S' from dad and a `s' from mom and that suggests one out of four doesn't it? But don't forget, mom could also be making that `S', and dad could be providing the `s'. So I'm going to write that over here as `sS'. So what are the odds of getting a `S' paternal/maternal combination? Get it, father/mother combination? One out of two. What are the odds of getting the `sS' one out of two. But these don't seem to be the same ideas we had before because these are not, these are not happening independently of each other, are they? I'm saying to you, you could get this `or' this, not `and.' You see? And so these two events, when I say to you what are the odds of getting `Ss'? Well, we could get a `Ss' by this paternal/maternal combination or the maternal/paternal combination, and all of a sudden it's not `and' anymore. It's `or.' We have two events that are sort of exclusive of each other.
And that brings us to our next law of probability, and that law is called the Additive Law, and again it's something many of you can intuit, but we better put it down in writing anyway; the Additive Law. These are just laws of probability. They're not necessarily Mendel's laws. Mendel used them, but they're laws of probability, and here it is. It's this: that the probability of one or another of two mutually exclusive events--in other words, that's almost a redundancy. If it's `or,' they're going to exclude each other, okay--of two mutually exclusively events occurring, equals what? The sum of their probabilities, of their individual probabilities. Again, intuition--a lot of probability is intuition. And if it's not intuition, just remember, okay, and it'll eventually become intuitive to you.
Let me show you what I mean. Let's go back to that last problem and see how that's going to work in terms of probability. We have a `Ss' crossed with a `Ss'. I want to know the probability of getting a `Ss'. Okay. I want to know the probability of getting a heterozygote. So we'll put our offspring right here. Well, now we can see. To get a `Ss' we have two different combinations. We could have the possible combinations are `Ss' or `sS' because typically they're going to be expressed the same way. So once again, for this one, what are the odds of getting--and we'll do it just the way I've been talking you through it. What are the odds of getting--we'll use dad first--I wonder if that's a male thing I got going. Anyway, we'll use the dad thing first. So dad, let's see. That's a `S', so what are the odds of getting an `S' from dad; one out of two, and what are the odds of getting a `s' from mom; now, once again, so this seems to be--what are the odds of getting a `S' from dad and getting a `s' from mom. See, they're' not mutually exclusive. They're happening independently of each other so that's our old law. That's the law of multiplication or the Multiplicative Law. So the odds of this combination with this coming from the father and this coming from the mother are one out of four. What are the odds of getting this combination with this coming from the mother and this coming from the father, the paternal? Well, once again, same thing; one out of two times one out of two equals one out of four.
So now though we have--this is an `or.' We're either going to have this or this. So the odds of this happening becomes the addition, the sum; two out of four which is equal to one-half, which goes along with your Punnett square. So you're saying once again, "why bother with Punnett squares?" And I'm saying but it can get ugly guys. All right?
And let's take a look at a typical di-hybrid cross. Now this is very interesting. Let's look at Mendel's 9 to 3 to 3 to 1 ratio. And you're saying but you know, after awhile this starts to break down. It doesn't make sense to me, but I want to show you something. Here is a chart that's kind of color coded to show you some very interesting things. We can look at this statistically and do the same kinds of things if we realize something, and I'll show you how to do it with gametes later, but I want you to realize something that these two traits, because they behaved independently of each other, literally follow the same rules as a monohybrid cross. Let me show you what I mean. 9 to 3 to 3 to 1--let's remember what that meant. The 9 was what showed both dominant traits. So all the yellow smooths here are literally, or rounds, are literally showing both dominant traits, but I want you to understand something. Look at these--one, two, three. You have literally 12 yellows. So if you consider yellow by itself--forget green for a second, or excuse me, forget wrinkles for a second--let's just consider yellow. If I said to you, if you crossed two yellow heterozygotes, what would you expect? Here's what you would expect; `Yy' crossed with `Yy,' what would you expect? You would expect yellow and , what? You would expect green. Forget the wrinkles for a second and look what we have. If I had 16 of these, if I had 16 of those, how many would you expect to be yellow? You would expect 12 out of 16, wouldn't you? Because yellow, if you had 16 you'd expect 12 out of 16. Look you guys. 9 to 3. Let's consider this the one with yellow. You have 12 yellows--1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 yellows. So, in other words, Mendel realized this. He was a statistician. He realized that they're just behaving as if they're independently assorted. Well, what would you expect with green? It's the same thing. Or smooth, excuse me. With the smooth, or the round as we called it before, okay, it's the same kind of thing. Look, all these are round, okay, everything up here, and 1, 2, 3. So we have 9 plus 3, 12 out of 16 are round. How cool is that. Do you get it? And if you get that, let me ask you to make a prediction. If you expect to be green, and you expect to be wrinkled, ooh, look at this. We have 1 out of 16--holy mackerel--which is exactly what we got. So now we can start using these ideas to work with di-hybrid crosses and the Additive Law.
For example, I'll do one real quick. All right, and then we'll do some extension lessons. All right. Let's do one real quick. What are the odds of getting, from this cross, `Yy'--we'll do S's and Y's--`Ss,' `Yy'; what are the odds of getting a yellow wrinkled? You know the answer right; 3/16, but don't tell anyone I told you that, okay, but let's take a look at that. What are the possible genotypes we need to get a yellow wrinkled? Let's list them. Here are the possible genotypes. To get a yellow wrinkled you could have `Yy,' `ss.' To get that you could also have `YY,' `ss,' and you could have, because we're crossing these, right, we're doing a di-hybrid cross, you could also have a `yY,' `ss.' You with me? So, therefore, look, maternal first, paternal second; paternal first, maternal second; and, of course, the dominant.
Let's move this to the next page and finish this up. Now what do we have? I'm going to keep this right here so I don't lose track of what I was doing. We have `Yy,' `Ss' crossed with `Yy,' `Ss.' I'm saying to you we have to look for these combinations; `ss,' `Yy,' `ss,' and `YY,' `ss'; let's do the probabilities and add them together. Ready? What are the odds of getting a `y' from dad? All right. 1 out of 2. What are the odds of getting a `Y' from mom? Right. 1 out of 2. What are the odds of getting a `s' from dad? 1 out of 2. What are the odds of getting a `s' from mom? 1 out of 2. What are the odds of getting this combination? 1/16. Do I need to work it through better? 1 out of 2, 1 out of 2, 1 out of 2, 1 out of 2. Right? `Y' from dad, `y' from mom, `s' from dad, `s' from mom. 1 out of 16. Look at this. What are the odds of getting a `Y' from dad? 1 out of 2. What are the odds of getting a `Y' from mom? 1 out of 2. Okay? We don't do a reciprocal situation here because it doesn't matter, but what are the odds of getting a `s' from mom, `s' from dad; 1 out of 16. What are the odds of getting this or this or this? 3 out of 16.
All right. Now look, I hope you understood that whole Additive Law with that di-hybrid cross because, remember, those things were mutually exclusive of each other and we had to go with the `or' statement there. But you also think back that I just showed you a little shortcut on how to deal with those di-hybrid crosses. And as long as you understand that mutually exclusive `or' thing, we can work with the short cut. But you gotta promise me you're going to make sure you understand that because it would break my heart if I felt like you weren't properly prepared to understand genetics, but let's go back and do that problem one more time with the shortcut.
So what did we have? We had `Yy,' `Ss' crossed with a `Yy,' `Ss' and I asked you what are the odds of getting a yellow wrinkled. All right. Now remember the short cut? The short cut was if you just look at these as separate crosses--so let's take a look at these color genes and treat them independently of the wrinkled genes and let's see what we're talking about here. So I'm going to take a `Yy' crossed with a `Yy' and look at my question. And my question was--what are the odds of getting a yellow? Well, you can do the Punnett square yourself, but I think by now you can do this. The odds of getting a yellow out of this are equal to 3 out of 4. All right. Now what are the odds of getting a wrinkled from my `S' crosses? Well, if I take `Ss' crossed with a `Ss,' what are the odds of getting a wrinkled, in other words, a `ss?' That's clearly 1 out of 4. So now we're--notice, we shortcutted this because we don't have to look at all the possible combinations, and now we are saying, what are the odds of getting a yellow and a wrinkled and a wrinkled? Multiplicative Law, times , 3 out of 16, that's cool.
So you see what we're talking about here. If you're familiar with the laws, with Mendel's laws and the way assortment happens in genes, there's a thousand different ways, well, two or three different ways, to skin a genetics problem. Good luck with these.
Mendelian Genetics and Mutation
Laws of Probability
Laws of Probability: The Additive Rule Page [1 of 2]