Laws of Addition and Multiplication

When considering the probability of two different events or outcomes, it is essential to clarify whether they are mutually exclusive or independent. If the events are mutually exclusive, then the probability that either one or the other will occur equals the sum of their individual probabilities. This is known as the law of addition.

If, however, two or more events or outcomes are independent, then the probability that both the first and the second will occur equals the product of their individual probabilities. This is known as the law of multiplication.

As a simple illustration of these laws, consider parents who have embarked upon their first pregnancy. The probability that the baby will be either a boy or a girl equals 1—i.e., 1/2 + 1/2. If the mother is found on ultrasonography to be carrying twins who are non-identical, then the probability that both the first and the second twin will be boys equals 1/4—i.e., 1/2 × 1/2.

Bayes’ Theorem

Bayes’ theorem, which was first devised by the Reverend Thomas Bayes (1702–1761) and published after his death in 1763, is widely used in genetic counseling. Essentially it provides a very valuable method for determining the overall probability of an event or outcome, such as carrier status, by considering all initial possibilities (e.g., carrier or non-carrier) and then modifying or ‘conditioning’ these by incorporating information, such as test results or pedigree information, that indicates which is the more likely. Thus, the theorem combines the probability that an event will occur with the probability that it will not occur. The theorem lay fairly dormant for a long time, but has been enthusiastically employed by geneticists. In recent years its beauty, simplicity and usefulness have been recognized in many other fields—for example, legal work, computing, and statistical analysis—such that it has truly come of age.

The initial probability of each event is known as its prior probability, and is based on ancestral or anterior information. The observations that modify these prior probabilities allow conditional probabilities to be determined. In genetic counseling these are usually based on numbers of offspring and/or the results of tests. This is posterior information. The resulting probability for each event or outcome is known as its joint probability. The final probability for each event is known as its posterior or relative probability and is obtained by dividing the joint probability for that event by the sum of all the joint probabilities.

This is not an easy concept to grasp! To try to make it a little more comprehensible, consider a pedigree with two males, I₃ and II₁, who have a sex-linked recessive disorder (Figure 22.1). The sister, II₂, of one of these men wishes to know the probability that she is a carrier. Her mother, I₂, must be a carrier because she has both an affected brother and an affected son (i.e., she is an obligate carrier). Therefore, the prior probability that II₂ is a carrier equals 1/2. Similarly, the prior probability that II₂ is not a carrier equals 1/2.

FIGURE 22.1 Pedigree showing sex-linked recessive inheritance. When calculating the probability that II₂ is a carrier, it is necessary to take into account her three unaffected sons.

The fact that II₂ already has three healthy sons must be taken into consideration, as intuitively this makes it rather unlikely that she is a carrier. Bayes’ theorem provides a way to quantify this intuition. These three healthy sons provide posterior information. The conditional probability that II₂ will have three healthy sons if she is a carrier is 1/2 × 1/2 × 1/2, which equals 1/8. These values are multiplied as they are independent events, in that the health of one son is not influenced by the health of his brother(s). The conditional probability that II₂ will have three healthy sons if she is not a carrier equals 1.

This information is now incorporated into a bayesian calculation (Table 22.1). From this table, the posterior probability that II₂ is a carrier equals 1/16/(1/16 + 1/2), which reduces to 1/9. Similarly the posterior probability that II₂ is not a carrier equals 1/2/(1/16 + 1/2), which reduces to 8/9. Another way to obtain these results is to consider that the odds for II₂ being a carrier versus not being a carrier are 1/16 to 1/2 (i.e., 1 to 8, which equals 1 in 9). Thus, by taking into account the fact that II₂ has three healthy sons, we have been able to reduce her risk of being a carrier from 1 in 2 to 1 in 9.

Table 22.1 Bayesian Calculation for II₂ in Figure 22.1

Perhaps by now the use of Bayes’ theorem will be a little clearer. Try to remember that the basic approach is to draw up a table showing all of the possibilities (e.g., carrier, not a carrier), then establish the background (prior) risk for each possibility, next determine the chance (conditional possibility) that certain observed events (e.g., healthy children) would have happened if each possibility were true, then work out the combined (joint) likelihood for each possibility, and finally weigh up each of the joint probabilities to calculate the exact (posterior) probability for each of the original possibilities. If this is still confusing, some of the following examples may bring more clarity.

Reduced Penetrance

A disorder is said to show reduced penetrance when it has clearly been demonstrated that individuals who must possess the abnormal gene, who by pedigree analysis must be obligate heterozygotes, show absolutely no manifestations of the condition. For example, if someone who was completely unaffected had both a parent and a child with the same autosomal dominant disorder, this would be an example of non-penetrance. Penetrance is usually quoted as a percentage (e.g., 80%) or as a proportion of 1 (e.g., 0.8). This would imply that 80% of all heterozygotes express the condition in some way.

For a condition showing reduced penetrance, the risk that the child of an affected individual will be affected equals 1/2—i.e., the probability that the child will inherit the mutant allele, × P, the proportion of heterozygotes who are affected. Therefore, for a disorder such as hereditary retinoblastoma, an embryonic eye tumor (p. 215), which shows dominant inheritance in some families with a penetrance of P = 0.8, the risk that the child of an affected parent will develop a tumor equals 1/2 × 0.8, which equals 0.4.

A more difficult calculation arises when a risk is sought for the future child of someone who is healthy but whose parent has, or had, an autosomal dominant disorder showing reduced penetrance (Figure 22.2).

FIGURE 22.2 I₂ has an autosomal dominant disorder that shows reduced penetrance. The probability that III₁ will be affected has to take into account the possibility that his mother (II₁) is a non-penetrant heterozygote.

Let us assume that the penetrance, P, equals 0.8. Calculation of the risk that III₁ will be affected can be approached in two ways. The first simply involves a little logic. The second uses Bayes’ theorem.

FIGURE 22.3 Expected genotypes and phenotypes in 10 children born to an individual with an autosomal dominant disorder with penetrance equal to 0.8.

Table 22.2 Bayesian Calculation for II₁ in Figure 22.2

The posterior probability that II₁ is a heterozygote equals 1/2(1 − P)/[1/2(1 − P) + 1/2], which reduces to {1 − P/2 − P}. Therefore, the risk that III₁ will both inherit the mutant allele and be affected equals ({1 − P/2 − P}) × 1/2 × P, which reduces to {(P − P²)/(4 − 2P)}. If P equals 0.8, this expression equals 1/15 or 0.067.

By substituting different values of P in the above expression, it can be shown that the maximum risk for III₁