# Using Probability to Solve Complex Genetics Problems

## Coins and Segregation

Let's say you have two coins. One is a penny and the other is a quarter. The flip of each coin can get you a pair of possible outcomes, heads or tails. If you flip the penny and get heads, does that result have any bearing on whether or not you'll get a heads or tails when you flip the quarter next? No! That's because these are two independent events.

Instead of flipping coins, when it comes to genetics, when each allele pair segregates during gamete (sex cell) formation, they do so independently as well, according to the **Mendel's second law**, the law of independent assortment. This means that a multi-character cross is like two or more independent monohybrid crosses that occur at the same time.

Consequently, we can figure out the probability that specific genotypes will occur in the F2 generation using the concepts we're going to go over in this lesson.

## Probability Images

Using the image on your screen (see video), you can see that we have constructed a dihybrid cross between two *RrYy* F1 heterozygotes. These were derived from a parental generation using a cross between an *RRYY* and an *rryy*. The male gametes of the F1 generation, thus turn out to be *RrYy* and the female gametes of the F1 generation are also *RrYy*.

*R* refers to the dominant allele for seed shape, round. Little *r* refers to the recessive allele for seed shape, wrinkled. *Y* refers to the dominant allele for seed color, yellow, and little *y* refers to the recessive allele for seed color, green.

The image has constructed a square to help us figure out the probability of any genotype that may result in the F2 generation. For instance, we can see that, in total, we can have 16 different possible outcomes, ranging from *RRYY* to *rryy*.

If I were to ask you, what is the probability that the F2 offspring will be *RRYY*? You'd look at the square and tell me that it would be 1/16, because only one such possibility exists. If I were to ask you, what is the probability that the offspring are *RRYy*? You'd tell me it is 1/8, because there are two such possible outcomes, and 2/16 = 1/8.

## Probability Math

Now, how can we use math to more quickly figure all of this out without having to construct a complex image of 16 possible outcomes or look through the square carefully to ensure we didn't miss the genotype in question?

In this case, let's simplify everything to a Punnett square showing us a monohybrid cross of *Yy* plants and another Punnett square showing us a monohybrid cross of *Rr* plants (see video). We can see that for the monohybrid cross of *Yy* plants, the probabilities of the different offspring genotypes are ¼ for *YY*, ½ for *Yy*, and ¼ for *yy*. Similarly, using the monohybrid cross for *Rr* plants, the probabilities of the different offspring genotypes are ¼ for dominant (*RR*), ½ for heterozygous (*Rr*'), and ¼ for recessive (*rr*).

That's pretty easy, since we only had to construct a square with four possible outcomes to get our probabilities. Actually, you didn't even have to construct the square to figure all of this out if you simply used the multiplication rule with respect to Mendelian inheritance, as outlined in another lesson. The **multiplication rule** is the multiplication of the probability of one event by the probability of the other event.

Actually, we're going to use this same rule to solve these even more complex genetic problems.

Let's say I ask you to tell me what the probability of *RRYY* is, without constructing a complex square as we did in the last section. How would you figure this out? Well, we now know that the probability of *RR* is ¼ and the probability of *YY* is also ¼. Using the multiplication rule, we multiply these two probabilities together to find the probability of *RRYY*. ¼ * ¼ = 1/16, the same answer as in the last section!

Good. Now tell me what is the probability of getting *RRYy* in the F2 offspring? The probability of *RR* is 1/4 while the probability of *Yy* is 1/2. 1/4 * 1/2 = 1/8. Again, the math confirms our visual image from the prior section.

Okay, one more example. What is the probability of getting an *rrYY*? It's simply ¼ * ¼ = 1/16.

## Lesson Summary

Now you know how to solve more complex genetic problems, either using visual images or math, thanks to **Mendel's second law** (the law of independent assortment) and the multiplication rule. The **multiplication rule** is the multiplication of the probability of one event by the probability of the other event. By using the math we went over in this lesson, you can easily solve similar problems on your own!

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