Binomial Equation (Lecture 5)

Question 1

Human families are relatively small, how does this affect phenotypic ratios?

Answer 1

Human families are relatively small, therefore phenotypic ratios among offspring often deviate significantly from Mendelian expectations.

Question 2

Consider a couple, each heterozygous (Cc) for a recessive allele that causes a serious disease in homozygous individuals.

If they have four children, what is the probability that exactly one is affected?

Does it matter that we typically only have one child at a time, not all four at once?

Answer 2

A=Affected

U=Unaffected

P(A) = P(cc) = 1⁄2 x 1⁄2 = 1⁄4

P(U) = P(CC) + P(Cc) = 1⁄4 + 1⁄2 = 3⁄4

This is identical to Mendel’s pea experiments (dominant versus recessive), but we must take into account that each birth is independent. So the first, second, third or the fourth child could be affected! We also have consider the possibility that none of the children could be affected or that all children could be affected, or 2 or 3 children could be affected. Whew! That means there are 5 possible outcomes!

Question 3

Consider a couple, each heterozygous (Cc) for a recessive allele that causes a serious disease in homozygous individuals.

Which Child will be affected? First determine how many outcomes there are

Answer 3

First, let’s look at all of the 5 possibilities (U=unaffected, A=affected):

1. 0 are affected: 4U, 0A: UUUU
2. 1 is affected: 3U, 1A: AUUU, UAUU, UUAU, UUUA
3. 2 are affected: 2U, 2A: UUAA, UAAU, AAUU, UAUA, AUAU, AUUA
4. 3 are affected: 1U, 3A: UAAA, AUAA, AAUA, AAAU
5. 4 are affected: 0U, 4A: AAAA

We are interested in scenario #2: 1 is affected

Therefore, there are 4 possible outcomes!

Question 4

1. 0 are affected: 4U, 0A: UUUU
2. 1 is affected: 3U, 1A: AUUU, UAUU, UUAU, UUUA
3. 2 are affected: 2U, 2A: UUAA, UAAU, AAUU, UAUA, AUAU, AUUA
4. 3 are affected: 1U, 3A: UAAA, AUAA, AAUA, AAAU
5. 4 are affected: 0U, 4A: AAAA

Can we describe these different scenarios mathematically?

Answer 4

Yes, using factorials

Question 5

What is the binomial equation and what does each part represent?

Answer 5

[n!/x! X y!]

n= the total number of children

x= the number of unaffected

Y= number of affected

!= all and including (ex. if n=4 then n! = 4x3x2x1)

Question 6

Given this image what is the probability of unaffected and affected?

(Hint: use math)

Answer 6

3/4 x 3/4 x 3/4. x 1/4

3 kids being Unaffected (CC or Cc). 1 kid being Affected (cc) Let’s call the Unaffected kids ‘x’ Lets call the Affected and the probability of them being child ‘y’ and his/her unaffected ‘p’. Here x=3, p=3/4. probability ‘q’. Here y= 1 Probability = (3⁄4)^3 and q= 1/4. Prob is (1/4)^1

Question 7

Mathematically, for a total number of n progeny, we can calculate the binomial probability that exactly x progeny will fall into one class, and y into the other class as:

[n!/x! y!]p^x q^y

with probabilities of occurrence of p and q (and p+q=1)

Using this what is the solution to the problem:

Answer 7

Thus, if you had 4 kids, the probability of having 3 unaffected and 1 affected would be 42%. Recall simple Mendelian ratios, if you had all the children at once there would be a 25% probability that one is affected, but this doesn’t typically happen in human births

Question 8

What is the binomial probability?

Answer 8