### 12.9: Probability Laws

#### Overview

The probability of inheriting a trait can be calculated using the sum and product rules. The sum rule is used to calculate the probability of mutually exclusive events. The product rule predicts the probability of multiple independent events. These probability rules determine theoretical probability—the likelihood of events occurring before they happen. Empirical probability, by contrast, is calculated based on events that have already occurred.

#### The Sum and Product Rules Are Leveraged to Calculate Inheritance Probabilities

Although Punnett squares are useful for visualizing the inheritance of one or two traits, they become cumbersome when applied to more complex scenarios. A Punnett square displaying just three traits contains 64 possible crosses. Probability laws enable much more efficient calculations of trait inheritance probabilities.

Consider a pregnant woman who wants to understand her child’s risk of inheriting biotinidase deficiency (BTD), an autosomal recessive disease that runs in her family. Infants with untreated BTD exhibit developmental delays, poor muscle tone, skin rashes, and hair loss. Severe cases are associated with seizures and loss of vision and hearing, among other symptoms. Neither the woman nor her parents have BTD, but her brother is affected, meaning that both parents must have one causal gene variant (i.e., both parents are heterozygotes, or carriers).

The probability of the woman’s child inheriting BTD is contingent on whether the woman is a carrier, possessing one causal allele. Since BTD is autosomal recessive and she is unaffected, she cannot have two causal alleles. However, she may carry one causal allele that could be passed to her child.

The first step toward ascertaining the child’s risk is to determine the likelihood of the mother being a carrier. This is achieved using the sum rule of probability. The sum rule states that the probability of mutually exclusive events is the sum of their individual probabilities. In this case, mutually exclusive events are possible parental allele combinations the pregnant woman may have inherited. Since both of her parents are heterozygotes (*Bb* genotype), she has one of four possible genotypes: paternal *B* and maternal *B* (*BB*), paternal *B* and maternal *b* (*Bb*), maternal *B* and paternal *b* (*Bb*), or paternal *b* and maternal *b* (*bb*). Since she doesn’t have BTD, the *bb *genotype can be ruled out. There are thus three possible genotypes with equal probabilities of 1/3, and two of these result in being a carrier (*Bb*). Hence, according to the sum rule, her probability of being a carrier is 2/3 (1/3 + 1/3).

Another critical factor in the child’s risk of BTD is the father’s probability of being a carrier. Here, the product rule comes into play. The product rule states that the probability of multiple independent events is the product of the events’ individual probabilities. If the mother is a carrier, this does not influence whether the father is a carrier. Thus, they are independent events.

For the child to inherit BTD, multiple independent events must occur. First, the mother must be a carrier (2/3 probability). Second, the father must be a carrier. If the father is unaffected and has no family history of BTD, his probability of being a carrier is considered equivalent to that of the general population (1/120). Third, the child must inherit the recessive allele from both parents (1/4 probability if both parents are carriers). According to the product rule, the child’s risk of inheriting BTD is the product of each of these probabilities: (2/3) x (1/120) x (1/4) = ~0.0014, or about 0.14%.

#### Theoretical and Empirical Probabilities Are Respectively Calculated before and after Events

The child’s 0.14% risk of BTD, a prediction calculated before birth, is a theoretical probability. It is possible, however, that the child and his or her siblings will inherit BTD, representing an empirical probability of 100%. Unlike theoretical probabilities, which are calculated before events have occurred, empirical probabilities are based on observations. When analyzing single pedigrees, theoretical and empirical probabilities may be very different. However, as more pedigrees are analyzed, theoretical and empirical probabilities become increasingly aligned.

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