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The third law, assuming and independent, and conditional on is

(1.10)

Notice that the event appears as a conditioning event in all the probability expressions. The laws are the same as before but with this simple extension.

Consider Table 1.3 again. If DVI and DAI were independent then the probability of both occurring in a road accident fatality would be the product of the probability of each happening separately. Thus

However, it is not the case that 9.4% of road accident fatalities have both injuries. An examination of Table 1.3 illustrates that this is not so. From Table 1.3 it can be seen that 14/120 = 0.12 or 12% of fatalities have both injuries. In such a situation where it can be said that DVI and DAI are not independent.

As another example of the use of the ideas of independence, consider a diallelic system in genetics in which the alleles are denoted and , with . This gives rise to three genotypes that, assuming Hardy–Weinberg equilibrium to hold, are expected to have the following probabilities

  (homozygotes for allele ),

  (heterozygotes),

  (homozygotes for allele ).

The genotype probabilities are calculated by simply multiplying the two allele probabilities together on the assumption that the allele inherited from one's father is independent of the allele inherited from one's mother. The factor 2 arises in the heterozygous case because two cases must be considered, that in which allele was contributed by the mother and allele by the father, and vice versa. Both of these cases have probability because of the assumption of independence (see Table 1.4). Note that The particular locus under consideration is said to be in Hardy–Weinberg equilibrium when the two parental alleles are considered as independent.

Table 1.4 Genotype probabilities, assuming Hardy–Weinberg equilibrium, for a diallelic system with allele probabilities and .

Allele from mother	Allele from father

This law may be generalised to more than two events. Consider events . If they are mutually independent then