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      Independent events are exchangeable. It is not necessarily the case that exchangeable events are independent. See Taroni et al. (2018) for a discussion. Also, two events which are mutually exclusive cannot be independent. As an example of independence, consider the rolling of two six‐sided fair dice, and say. The outcome of the throw of does not affect the outcome of the throw of . If lands 6 uppermost, this result does not alter the probability that will land 6 uppermost. The same argument applies if one die is rolled two or more times. Outcomes of earlier throws do not affect the outcomes of later throws. Similarly, with the drawing of two cards from a pack of 52 cards, if the first card drawn is replaced in the pack, and the pack shuffled, before the second draw, the outcomes of the two draws are independent. The probability of drawing two aces is 4/52 4/52. This can be compared with the probability 4/52 3/51 if the first card drawn was not replaced.

       Third law of probability for independent events

      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 .

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Allele from mother	Allele from father