Question

Advanced Properties of Probability. (20 points total) Suppose you are culturing cells. You have three colonies each in its own T-75 cell culture flask, and each with its own genotype. The flasks are designated as: Flask1: homozygous dominant, (Le, alleles AA) Flask 2: homozygous recessive (i.e., alleles aa) Flask 3: heterozygous dominant (i.e., alleles Aa) To pass the time, your labmate decides to play a bit of a guessing game. He explains that he will randomly select a flask, and then randomly reveal one of the alleles of the genotype in the flask. You are to then guess the other allele of the genotype. (a) He tells you the allele he has selected is dominant (i.e., A). What is the probability that the other allele is also dominant? (Hint: it is not 1/2.) (b) Based on your answer, what is your strategy to win the game?

Answer 1

(a)

Let A and a be the event that the selected allele is A or a.

Let F1, F2 and F3 be the event that the flask 1, 2, or 3 is selected.

Then,

P(A | F1) = 1 , P(a | F2) = 1, P(A | F3) = 1/2, P(a | F3) = 1/2

As, he has randomly selected the flask, probability of selecting any flask, P(F1) = P(F2) = P(F3) = 1/3

By law of total probability,

P(A) = P(F1) * P(A | F1) + P(F3) * P(A | F3)

= (1/3) * 1 + (1/3) * (1/2) = 1/2

Given, that the selected allele is dominant, Probability that the other allele is also dominant = Given, that the selected allele is dominant, Probability that Flask 1 was selected

= (F1 | A) = P(A | F1) * P(F1) / P(A) (By Bayes theorem)

= 1 * (1/3) / (1/2) = 2/3

(b)

In part (a), we have determined if the selected allele is A, then the probability that other allele is A is 2/3.

Now,

By law of total probability,

P(a) = P(F2) * P(a | F2) + P(F3) * P(a | F3)

= (1/3) * 1 + (1/3) * (1/2) = 1/2

Given, that the selected allele is a, Probability that the other allele is also a = Given, that the selected allele is dominant, Probability that Flask 2 was selected

= (F2 | a) = P(a | F2) * P(F2) / P(a) (By Bayes theorem)

= 1 * (1/3) / (1/2) = 2/3

Thus, if the selected allele is a, then the probability that other allele is a is 2/3.

The strategy to win the game is,

If the selected allele is A, guess the second allele as A and

If the selected allele is a, guess the second allele as a.