Binomial Probability and Combinatorics

The binomial distribution calculates the probability of a specific number of successes (e.g., affected children) in a fixed number of independent trials (e.g., pregnancies), given a constant probability of success. This commonly arises in recurrence-risk counseling.

P(k affected out of n children) = C(n,k) x p^k x (1-p)^(n-k)

Where:

• C(n,k) = n! / (k! x (n-k)!): the number of ways to choose k from n
• p = probability of affected in each pregnancy
• k = number of affected children
• n = total number of children

Example 1: Exactly 2 out of 4 affected

Problem: Two carrier parents (AR condition, p=1/4 per child). What is the probability that exactly 2 out of their 4 children are affected?

Solution:

• C(4,2) = 4! / (2! x 2!) = 6
• P = 6 x (1/4)^2 x (3/4)^2
• P = 6 x 1/16 x 9/16
• P = 54/256 = 27/128 (approximately 21%)

Example 2: At least 1 affected

Problem: Same couple, 3 children. What is the probability that at least 1 is affected?

Solution (complement method):

• P(at least 1) = 1 - P(none affected)
• P(none) = (3/4)^3 = 27/64
• P(at least 1) = 1 - 27/64 = 37/64 (approximately 58%)

Example 3: All unaffected

Problem: Same couple, 5 children. What is the probability all 5 are unaffected?

Solution:

• P(all unaffected) = (3/4)^5 = 243/1024 = approximately 24%

For independent events, the probability of all events occurring is the product of their individual probabilities.

Maximum number of haplotypes: For a gene with n heterozygous loci, the maximum number of distinct haplotypes = 2^n.

Example: If a person is heterozygous at 3 loci on the same chromosome (A/a, B/b, C/c), there are 2^3 = 8 possible haplotype combinations.

Always use the complement:

• P(at least 1) = 1 - P(none)
• This is much easier than calculating P(exactly 1) + P(exactly 2) + ... + P(all)

• Each pregnancy is independent: Previous outcomes do not change the probability for the next child. A couple with 3 unaffected children still has a 1/4 risk for the next child.
• "At least" vs "exactly": Read the question carefully. "At least 1" requires the complement method. "Exactly 2" requires the full binomial formula.
• Forgetting C(n,k): The binomial coefficient accounts for the different orderings. Without it, you only calculate the probability of one specific ordering.