In this lesson, you’re going to learn some basics concepts related to probability and how to apply the addition rule to problems related to Mendelian inheritance.

## Cards ; Inheritance

If you’re a fan of card games like me, then you know a standard deck of cards contains 52 cards. What’s the probability that you’ll pull out a ten of hearts randomly from the deck? Well, that ten is the one and only ten of hearts in a deck of 52 cards. That means the probability of pulling that card out is 1/52.

That wasn’t too hard, was it? Using simple concepts of probability just like this, let’s find out how we can apply the addition rule to Mendelian inheritance.

## Probability Review

Before we get to the harder parts of this lesson, I need to make sure you know some very basic concepts related to probability. If you have a deck of 52 cards and all the cards are aces of spades, what is the probability that you’ll pull out an ace of spades? 1. If an event is certain to occur, it has a probability of 1.

Using this same exact deck, what’s the probability you’ll pull out an ace of hearts? 0, because the deck is stacked solely with aces of spades. If an event is certain to not occur, then it has a probability of 0.

Now let’s switch back to a normal, standard deck of 52 cards. You know that the probability you’ll pull out a two of diamonds is 1/52, a three of diamonds is 1/52, a ten of spades is 1/52, and so on for every one of the different cards in the 52 card deck. The probabilities of all the possible outcomes for an event have to add up to 1. The probabilities 1/52 + 1/52 + 1/52; for all the possible outcomes (the different cards you can pull out) will add up to 1.

Now, let’s turn our attention to a couple of coins. One is a penny, and the other is a nickel. You’re probably aware that you can flip each coin onto its head or tail. You also know that the outcome of each coin’s flip is independent of the outcome of the other coin’s flip, whether it occurs simultaneously or not. This means that each coin toss is an independent event because the outcome of any toss of any coin is independent of the outcome of any of its prior tosses or simultaneous tosses of another coin.

This is similar to **Mendel’s second law**, the law of independent assortment, which boils down to the fact that the alleles of one gene segregate into gametes independently of the alleles of another gene. This means that if you flip a penny and get tails, it won’t influence the outcome of flipping the nickel. Except with respect to genetics, we’re talking about a pair of alleles of a gene, as opposed to a pair of sides of a coin.

## Addition Rule

Let’s apply your knowledge of probability to Mendelian inheritance using pea plants.

The P generation is the parental generation which gives rise to an F1, or first filial generation monohybrid cross in our example. The F1 hybrids have a genotype of *Aa*. When the F1 hybrids either self-pollinate or cross-pollinate with other F1 hybrids, they produce the F2 generation, the second filial generation, where a genotype of *aa* leads to wrinkled seeds.

As an example, to find out the probability that an F2 plant will be heterozygous (*Aa*) as opposed to homozygous (*aa* or *AA*), we need to first understand that the dominant allele, *A*, can come from the egg or sperm. The same holds true for the recessive allele, *a*. This means that F1 gametes (the sperm and egg) will combine to produce *Aa* offspring in two mutually exclusive ways.

The **addition rule** tells us that the probability that any one of two or more mutually exclusive events will occur can be ascertained by the addition of their individual probabilities. What do I mean by mutually exclusive? Let’s go back to our deck of cards and our coins. An event is mutually exclusive if it can’t occur at the same time. I can only flip a coin onto head or tails, it can’t possibly flip onto both at the same time!

On the other hand, a non-mutually exclusive event would be like taking a card out of a 52-card deck that is either red or a ten. These are not mutually exclusive events because you can pull out a red ten, can’t you? Of course! Meaning, non-mutually exclusive events can happen separately, you can pull out something like a red ace or a black ten, or it can happen at the same time, you can pull out a ten of diamonds.

Okay, back to Mendel.

Using the multiplication rule, you will have learned that the probability than an F2 heterozygote will arise is ; if the dominant allele comes from the sperm and the recessive allele from the egg. Likewise, if the recessive allele comes from the sperm and the dominant allele from the egg, the probability that an F2 heterozygote will arise is also ;.

Using the addition rule, the probability that any one of these two possibilities will arise is thus ; + ; = ;.

## Lesson Summary

That shouldn’t have been too hard, right? Remember some basic rules related to probability:

- If an event is certain to occur, it has a probability of 1
- If an event is certain to not occur, then it has a probability of 0
- The probabilities of all the possible outcomes for an event have to add up to 1

We can apply these and other rules of probability to **Mendel’s second law**, the law of independent assortment. Namely, in this lesson we applied the **addition rule**, which tells us that the probability that any of two or more mutually exclusive events will occur can be ascertained by the addition of their individual probabilities. This means we add the probabilities of two mutually exclusive events together in order to figure out the probability that either one of them, as opposed to a specific one, will occur.