Statistics for laboratory scientists

Homework problems for lecture 6

1. Consider two events, A and B.

   1. Suppose that A and B are mutually exclusive. Calculate, in terms of Pr(A) and Pr(B),

      1. Pr(A or B)

      2. Pr(A and B)

   2. Suppose that A and B are independent. Calculate, in terms of Pr(A) and Pr(B),

      1. Pr(A and B)

      2. Pr(A or B)

   3. Suppose that A and B are both mutually exclusive and independent. What can we say about Pr(A or B), Pr(A and B), Pr(A) and Pr(B).

2. Consider a rare recessive trait. Let p be the frequency of the disease allele, d. (Let + denote the normal allele. Assume Hardy-Weinberg equilibrium and random mating, so that the frequencies of the three genotypes, ++, +d, and dd are (1-p)², 2p(1-p), and p², respectively.

   Consider picking a woman at random from the population. Let U = {She is unaffected}, C = {She is a carrier}, A = {her brother is affected and her parents are both unaffected}, B = {her first child is affected}, D = {at least one of her five children is affected}. Note: let's assume that "carrier" means having either genotype +d or dd.

   Assume that p = 1%, and calculate the following.

   1. Pr(C | U)

   2. Pr(C | U and A)

   3. Pr(B | U and A)

   4. Pr(D | U and A)

3. Consider two urns. Urn A contains 5 green balls and 5 blue balls. Urn B contains 2 green balls and 8 blue balls. I roll a six-sided, fair die. If I get a 3 or a 4, I draw two balls without replacement from urn A; otherwise, I draw two balls without replacement from urn B.

   Let A = {draws are made from urn A}, B = {draws are made from urn B}, G₁ = {first ball is green} and G₂ = {second ball is green}.

   Calculate the following.

   1. Pr(G₁)

   2. Pr(G₂ | A and G₁)

   3. Pr(G₂ | A)

   4. Pr(G₂ | B and G₁)

   5. Pr(G₂ | G₁)

   6. Pr(exactly one green ball)

   7. Pr(A | at least one green ball)

4. Suppose that a mare is corralled with two stallions: a champion and a dud. The mare gets pregnant and produces a colt.

   1. What is Pr(the champion is the colt's father)?

   2. Suppose that the champion is known to carry a rare marker on its Y chromosome, present in only 2% of male horses. Suppose that the colt is found to also carry this Y marker. What is Pr(the champion is the colt's father | this information)?

   3. Suppose that, in addition to the conditions in (b), the mare had been "exposed" to another 998 stallions. What is Pr(the champion is the colt's father | this information)?