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.

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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.

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