7.6–7.8: Probability and chance play central roles in genetics.

Because of the role of chance in genetics, we cannot always predict the exact outcome of a genetic cross.

Sometimes genetics is a bit like gambling. Even with perfect information, it can still be impossible to know the genetic outcome with certainty. It’s like flipping a coin: you can know every last detail about the coin, but you still can’t know whether the coin will land on heads or tails. The best you can do is define the probability of each possible outcome.

The rules of probability (the same ones that govern coin tosses and the rolling of dice) have a central role in genetics, for two reasons. The first is a consequence of segregation, the process Mendel described, in which each gamete an individual produces receives only one of the two copies of each gene that the individual carries in most of its other cells. As a result, it is equally likely that the haploid gamete—the sperm or egg—will include one allele or the other allele of the two that the individual carries. It is impossible to know which allele it will be. The second reason probability plays a central role in genetics is that fertilization, too, is a chance event. All of the sperm or eggs produced by an individual are different from one another, and any one of those gametes may be the gamete involved in fertilization. Thus, knowing everything about the alleles a parent carries is not always enough to be able to determine with certainty which alleles his or her offspring will carry.

Let’s explore how we can make predictions in games of chance, as well as in matings, based on probabilities. If an individual is homozygous for a trait, 100% of his or her gametes will carry that allele. Any gamete produced by an individual heterozygous for a trait has a 50% probability of carrying the dominant allele and a 50% probability of carrying the recessive allele. Let’s use these probabilities in an example.

If a male is heterozygous for albinism (Mm) and a female is albino (mm), what is the probability that an offspring of theirs will be homozygous for albinism (mm)? To get the mm outcome, two events must occur. First, the father’s gamete must carry the recessive allele (m), and second, the mother’s gamete must carry the recessive allele (m). In this case, the probability of a homozygous recessive offspring is 0.5 (the probability that the father’s gamete carries m) times 1.0 (the probability that the mother’s gamete carries m), for a probability of 0.5. This is a general rule when determining the likelihood of a complex event occurring: if you know the probability of each component that must occur, you multiply all the probabilities together to get the overall probability of that complex event occurring (FIGURE 7-13).

Figure 7.13: Using probability to determine the chance of inheriting albinism.

Consider another example, involving Tay-Sachs disease. As described in Chapter 3, Tay-Sachs is caused by malfunctioning lysosomes that do not digest cellular waste properly. It leads to death in early childhood. Tay-Sachs occurs if a child inherits two recessive alleles for the Tay-Sachs gene. If each parent is heterozygous for the Tay-Sachs gene, what is the probability that their child will have Tay-Sachs disease?

To solve this, break down the event of the child having Tay-Sachs into two separate events: first, it must inherit a recessive allele, t, from its heterozygous father. Because half the father’s sperm will have the dominant T allele and half will have the t allele, the probability of the child inheriting the t allele is 0.5. Second, the child must also inherit a recessive allele from its heterozygous mother. Because half of the mother’s eggs will have the T allele, while the other half have the t allele, the probability of the child inheriting the t allele from its mother is also 0.5. So the overall probability of the child having Tay-Sachs disease is 0.5 × 0.5 = 0.25, or 1 in 4 (FIGURE 7-14).

Figure 7.14: The chance of inheriting Tay-Sachs disease. The probability that a child will inherit Tay-Sachs disease from two heterozygous parents is 25%.

Of course, if the couple has only one child, we can’t predict with certainty whether the child will have Tay-Sachs. The mathematical probability tells us, however, that if the couple had an infinite number of children, we could expect that one-fourth of them would have Tay-Sachs disease. This fact may not be much help for heterozygous couples trying to decide whether to have a child or not.