Sheldon Ross 10 : Example 2.9

Left-handedness by Chandra Sekhar Kounduri


Question: Suppose that a particular trait of a person (such as eye color or left handedness) is classified on the basis of one pair of genes and suppose that d represents a dominant gene and r a recessive gene. Thus a person with dd genes is pure dominance, one with rr is pure recessive, and one with rd is hybrid. The pure dominance and the hybrid are alike in appearance. Children receive one gene from each parent. If, with respect to a particular trait, two hybrid parents have a total of four children, what is the probability that exactly three of the four children have the outward appearance of the dominant gene? 


Step 01: Creation of the variables


Step 02: Possible Children

Since both of the parents are mixed, any of the child born to this set of parents would receive either “r” or “d” from either of the parents. The types of children possible from this union are as follows.

Amongst the progeny, the children “dd”, “dr”, “rd” would be outwardly display the dominant characteristics. This is ¾.


Step 03: Numerical Solution

According to the question, we need to find the probability that 3 out of 4 children would have the outward appearance of the dominant gene. This is a binomial type problem and below is the numerical answer.

4C3 (¾)3(¼)1 = 2764 = 0.421875


Step 04: Simulation

We can also do a simulation of the same to see how the probabilities match up to the results calculated numerically.

You can see that the proportions (and the probabilities) match up at high numbers with the ratio being roughly ¼.

If only 4 children are allowed, then the probability of 3 of four children having an outwardly appearance of the dominant gene is 2764 ~ 0.42 according to our calculation earlier. This is seen below. Notice that we are repeating the allowed trials. For example, the first chart in the set has a number 10 under it. This means that this is a distribution of probability of 3 dominant children in a population where 10 paris of parents are allowed to have 4 children each. Observe how the distribution gets narrower with larger population.

 


End of the post


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