Use the Binomial Rule

Michael writes the calculation on the whiteboard using the binomial rule: 

4 × (¼) × (¾)³ = 4 × ¼ × (27/64) = 108/256 = 42.2%

Dr Antoniou looks around, noticing a few puzzled expressions in the room. Some students seem unsure about the path from this numeric expression to a deeper understanding of how the binomial rule functions in genetics.

Dr Antoniou: ‘This formula emerges from the binomial expansion (a + b)ⁿ. For a single event, such as a child’s genotype, if ‘a’ is the probability of being affected (1/4) and ‘b’ is the probability of being unaffected (3/4), and if we have n = 4 children, then any specific number of affected children can be determined by the term:

p = k x ak x b(n-k)

where, 

•  p is the probability of the event we want to calculate (this case 1 in 4 children having the disease)

•  k is the number of events we want (this case 1 child)

•  n is the total number of events (this case 4 children)

•  a is the probability of the event we want (this case ¼)

•  b is the probability of the other events (this case ¾)

Since we wanted exactly one affected child among four, we set k = 1, a = 1/4, and b = 3/4. The coefficient (n choose k) = (4 choose 1) = 4 indicates the number of distinct ways to arrange exactly one affected child among four births. Each scenario has a probability of (1/4)¹ × (3/4)³. Multiplying these together:

4 × (1/4) × (3/4)³ = 4 × 0.25 × (0.75)³  = 0.421875, or about 42.2%.

This calculation accounts for each possible birth order of that single affected child.

We can easily adapt this approach for scenarios like ‘exactly two affected children out of four.’ In that case, we’d use (4 choose 2), (1/4)², and (3/4)². The crux is: the binomial theorem ensures we sum over all arrangements, rather than focusing on a single sequence of births. This is crucial in genetics, where every pregnancy is an independent event. By mastering this formula, you can predict the distribution of genotypes within families accurately and address questions about how many children might be affected or unaffected in varying circumstances.” 

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