Probability Rules

Probability rules are the building blocks for all genetic risk calculations. Before tackling Bayesian analysis or binomial problems, you need to be comfortable with these core rules.

The probability that two independent events both occur is the product of their individual probabilities.

P(A AND B) = P(A) x P(B)

This applies when events are independent: the outcome of one does not affect the other.

Genetics examples

• Risk of an affected child: If both parents are carriers (Aa x Aa), the chance of a child being aa = P(getting a from mom) x P(getting a from dad) = 1/2 x 1/2 = 1/4
• Compound probability across a pedigree: A woman has a 2/3 chance of being a carrier, her partner has a 1/25 chance. Risk of affected child = 2/3 x 1/25 x 1/4 = 1/150
• Two unaffected children in a row: (3/4) x (3/4) = 9/16

Maximum number of haplotypes

For n heterozygous loci on the same chromosome, the maximum number of distinct haplotypes = 2^n (multiplication rule applied to binary choices at each locus).

• 2 heterozygous loci: 2^2 = 4 haplotypes
• 3 heterozygous loci: 2^3 = 8 haplotypes
• 5 heterozygous loci: 2^5 = 32 haplotypes

The probability that at least one of two mutually exclusive events occurs is the sum of their individual probabilities.

P(A OR B) = P(A) + P(B) (when A and B cannot both happen)

If A and B are not mutually exclusive:

P(A OR B) = P(A) + P(B) - P(A AND B)

Genetics examples

• Carrier or affected: In a cross Aa x Aa, the chance a child is either a carrier OR affected = P(Aa) + P(aa) = 1/2 + 1/4 = 3/4
• More than one possible diagnosis: If a child has a 40% chance of condition A and a 30% chance of condition B (and the two are mutually exclusive), the probability of having either = 0.40 + 0.30 = 0.70

The probability that an event does NOT occur equals 1 minus the probability that it does.

P(NOT A) = 1 - P(A)

Genetics examples

• At least one affected child out of 4: P(at least 1) = 1 - P(none affected) = 1 - (3/4)^4 = 1 - 81/256 = 175/256 (approximately 68%)
• Not a carrier: If carrier risk is 2/3, then non-carrier risk = 1 - 2/3 = 1/3

This is by far the most efficient way to solve "at least one" problems. Never calculate P(exactly 1) + P(exactly 2) + ... when you can use the complement.

The probability of an event given that another event has occurred.

P(A | B) = P(A AND B) / P(B)

Genetics examples

• 2/3 carrier rule: P(carrier | unaffected) for offspring of two carriers. P(carrier AND unaffected) = 1/2. P(unaffected) = 3/4. So P(carrier | unaffected) = (1/2) / (3/4) = 2/3.
• Bayesian analysis is conditional probability: Every Bayesian calculation is asking "what is P(carrier | evidence)?"

Multi-step pedigree problem

Problem: A woman's maternal uncle has hemophilia A (X-linked recessive). She has 2 unaffected sons. What is the probability her next son will be affected?

Step 1: Determine her prior carrier risk:

• Her maternal grandmother is an obligate carrier
• Her mother has a 1/2 chance of being a carrier
• She has a 1/2 chance of inheriting from her mother (if her mother is a carrier)
• Prior = 1/2 x 1/2 = 1/4

Step 2: Update with 2 unaffected sons (Bayesian):

Step 3: Risk for next son:

• P(carrier) x P(affected son | carrier) = 1/13 x 1/2 = 1/26

This problem used: multiplication rule (step 1), conditional probability and Bayesian analysis (step 2), and multiplication rule again (step 3).

• Multiplying when you should add: "A OR B" is addition. "A AND B" is multiplication. Read the question carefully: "what is the chance the child is a carrier or affected?" vs "what is the chance both children are carriers?"
• Forgetting independence: The multiplication rule only works for independent events. Pregnancies are independent; alleles at linked loci are not.
• Skipping the complement shortcut: Anytime the question says "at least one," use 1 - P(none). It is always faster.
• Not conditioning properly: If you know an individual is unaffected, you must remove the affected possibility from the denominator (this is why carrier risk for an unaffected sibling is 2/3, not 1/2)