Question

Bob is an oncologist who focuses on the genetics of lab mice. He is researching a particular gene that has 29 different alleles (that is, there are 29 different variants of the gene). A mouse is equally likely to carry any of these 29 alleles.

(a) Suppose that Bob has 11 mice. He will run a genetic test to determine the allele of each separate  mouse. Let Y1 be the allele of mouse 1, Y2 be the allele of mouse 2, etc. An outcome of her test can be written as (Y1, Y2, ...., Y11). How many different possible outcomes are there? Side note to consider for the problem: Each of the 29 alleles is equally likely to occur in each mouse, so each outcome you counted in the previous part is equally likely.

(b) How many of the outcomes have 11 different alleles? Or how many outcomes of each(Y1, Y2, ..., Y11) don’t have an allele that appears more than once?

(c) What is the probability that none of the 11 mice will have the same allele? Answer to the nearest ten-thousandth

(d) What is probability that there will be at least two mice that have the same allele? Express your answer to the nearest ten-thousandth.

Answer 1

a) Each of the 11 mice can have any of the 29 alleles. So the total possible outcomes are: 29*29*....29 = 29^11 = 1.22*10^16

b) For the 11 mice to have different alleles, the number of ways are 29C11 = 34597290

c) Required probability = 29C11/29^11 = 2.836*10^-9

d) Probability that atleast two mice have the same allele = 1- P(0)- P(1) = 1- 2.836*10^-9 - 29C10*29/29^11 = 0.99999995