# Using the rules of probability to solve problems

The multiplication rule of probability states that the probability of one event **and **a second event **both** occurring is the product of the probabilities of each individual event occurring separately. This can be used in multiple ways in genetics.

# Predict the probability of offspring with a particular combination of traits

In the previous section, the multiplication rule was used to explain the 9:3:3:1 phenotypic ratio observed from a dihybrid cross. In a similar fashion, we can use it to predict the probability of offspring with a particular combination of traits, regardless of how many genes we track.

For example, let’s calculate the probability of an offspring with all recessive traits for the cross:

AaBbCc x AabbCc

We draw three separate Punnett squares, one for each gene, as shown in Figure 8. As seen by the Punnett squares, there is a ¼ probability of the recessive “a” phenotype. There is a ½ probability of the recessive “b” phenotype. And there is a ¼ probability of the recessive “c” phenotype. The probability of an offspring that has all three recessive traits is ¼ * ½ * ¼ = 1/32.

## Test Your Understanding

# Additional applications of the rules of probability

In previous sections, the product rule of probability was used to calculate the probability of obtaining offspring with one trait **and** another independently assorting trait. But the probability rule can be used in any circumstance to calculate the probability of two or more events **all **happening. For example, the probability rule could be used to calculate the probability that, in a monohybrid cross, two offspring in a row will be heterozygous. If the probability of one heterozygous offspring is ½, the probability of two in a row would be:

½ * ½ = ¼.

A non-biology example: if the probability of heads on a coin flip is ½, the probability of two heads in a row is ½ * ½ , the probability of three heads in a row is ½ * ½ * ½, and the probability of four heads in a row is:

½ * ½ * ½ * ½.

The multiplication rule is used to calculate probability of one thing **and** another **both** occurring.

The **addition rule of probability**, on the other hand, is used to calculate the probability of one thing **or** another occurring. For example, in a monohybrid cross, we’d expect ¼ AA offspring, ½ Aa offspring, and ¼ aa offspring. The probability of any single offspring having a genotype of AA **or** aa is ¼ + ¼ = ½.

We will see additional uses for both rules of probability in later chapters.

## Test Your Understanding

### Media Attributions

- Punnett squares © Amanda Simons is licensed under a CC BY-SA (Attribution ShareAlike) license