Mendelian Genetics 2Probability Theory and StatisticsMathematicians distinguish two kinds of processes:deterministic outcomes predicted exactly flip coin with two heads-> Hstochastic outcomes have probabilities flip coin with H and TModels in science:Deterministic Newtonian physics Stochastic quantum theoryIn everyday language, stochastic ≈ random; in probability theory, random sometimes restrictedto cases where all outcomes are equally probable. I’ll usually say “strictly random”.fair coin stochastic and (strictly) randomweighted coin stochastic, not strictly randomComputers generate pseudorandom numbers: start with number you give it or the time of day,go through long series of arithmetic operations. Resulting series of numbers could be predictedexactly and hence are determiistic IF you knew the starting number. If you don’t, almostimpossible to distinguish from strictly random.How Mendel saw randomness as variation among progeny of different plants.F2 from a cross showed 336 round:102 wrinkled, very close to 3:1. Stochastic or deterministic? Mendel looked at individual plants, got the following among others:round 45 43 14wrinkled 11 2 153:1 20:1 1:1Can use Punnett squares and intuition to solve manyproblems in genetics, i.e. to predict kinds andfrequencies of gametes and progeny, phenotypes andgenotypes. But can be very complicated; e.g. 3-factorcross may require up to 64-block Punnet square.Better to learn to use a bit of basic probability theory.Terminology:roll die P(6) = 1/6 = 0.1667 % Don't use %!! Usefraction or decimal fraction.probability of an event or outcome can range 0(impossible) --> 1 (must happen)P(r r --> R gamete) = 0P(r r --> r gamete) = 1P(R r --> R gamete) = 1/2Two “Kinds” of Probabilities, or Two Ways of Thinking About Them1. a priori probabilities are based on model or hypothesisE.g. toss coin. Whether lands H or T depends on details of how one flips it and where onecatches it. Assume we could never control thumb and hand precisely enough to controloutcome. Then either outcome equally likely, or P(H) = P(T) = 1/2. (I have read thatchaos theory has been used to verify this.)E.g. Mendel hypothesized that fusion of gametes is random with respect to the genes hestudied in peas. Self A a, P(A pollen & A egg) = P(A pollen & a egg), etc.2. a posteriori probabilities = observed frequencies of events"E.g. toss coin many times, frequency of heads = f(H) ≈ 1/2.E.g. Mendel observed frequencies A A = A a = a A = a aTo do most kinds of genetics, need learn only two basic probability rules and how to applythem.1. Independent events: Occurrence of one doesn't affect probability of the other.e.g. toss 2 coins or 1 coin 2 times, H1 and T2 are independentpick 1 egg and 1 pollen from Rr plant, R egg and R pollen are independentIf events M, N, O, ... are independent, P(M & N & O ... ) = P(M) P(N)P(O) ...e.g. toss 2 coins P(H1 & T2) = P(H1)P(T2) = (1/2)(1/2) = 1/4P(R egg and r pollen from Rr) = (1/2)(1/2) = 1/42. Mutually exclusive events: Cannot occur together.e.g. toss 1 coin, H and T are mutually exclusive, can get one or the other, not both1 gamete from Rr plant --> R or r, not bothIf events A, B, C ... are mutually exclusive, P(A or B or C ...) = P(A) + P(B) + P(C) ...e.g. P(H or T) = (1/2) + (1/2) = 1P(F2 from Rr X Rr is round) = P(RR or Rr) = P(RR) + P(Rr) = (1/4) + (1/2) = 3/4Same result as Punnet square and intuition.This is easy. Hard part:Know which rule to use.Know how to combine rules to solve problem.Do simple cases, relate to intuition and Punnett square.(1) Toss 2 coins. P(1 H & 1 T) = ? Order not specified, want any order.P(T,T) = P(H1 & T2 or T1 & H2) = [ P(H1)P(T2) ] + [ P(T1)P(H2) ] indep. indep. mutually exclusive (compound events)= (1/2)(1/2) + (1/2)(1/2) = (1/4) + (1/4) = 1/2cf. Punnett square1/4 H T + 1/4 H T = 1/2 H T1/4 T T1/4 T H1/4 H T1/4 H H Toss 21/2 H 1/2 T1/2 H1/2 TToss 1(2) Rr × Rr --> ?P(RR) = P(Rf & Rm) = (1/2)(1/2) = 1/4P(rr) = P(rf & rm) = (1/2)(1/2) = 1/4P(Rr) = P(Rf & rm or rf & Rm) = (1/2)(1/2) + (1/2)(1/2) = 1/2or P(Rr) = 1 – P(RR or rr) = 1 – [(1/4) + (1/4)] = 1/2The last point is very important; in many cases it is easier to calculate the probabilitythat something does not happen and subtract it from 1 than it is to calculate theprobability that it does happen directly.(3) Rr Yy Tt × Rr yy tt ---> 2,000 seeds How many do we expect to be round andgreen and produce tall plants?Translate to genotypes: expect how many R– yy T– ?Three pairs of alleles segregate independently, so start by doing each one separately.Rr × Rr --> 3/4 R–Yy × yy --> 1/2 yyTt × tt --> 1/2 TtP(R– yy T–) = (3/4)(1/2)(1/2) = 3/16 = expected frequencyexpected number = (3/16)(2,000) = 375Conditional ProbabilitiesConditional probabilities show how our assignment of probabilities depend on our prior knowledge.e.g. Rr X Rr --> 1/4 RR 1/2 Rr 1/4 rr What proportion of round peas are homozygous? Translateto probability language: what is the probability that a pea is homozygous, given that it is round?There is a law of conditional probabilities:P(A given B) = P(A and B)/P(B)P(A|B) = P(A∩B)/P(B)P(RR|R-) = P(RR∩R-)/P(R-) = (1/4)/(3/4) = 1/3But you don't have to use it in any situation that we will consider.Instead, note that when I specified that the peas must be round, I eliminated one possible outcome,wrinkled peas. This changes the probabilities:I have 2 children. What is P(2 F)?P(FF) = P(1stF & 2ndF) = P(1stF)P(2ndF) = (1/2)(1/2) = 1/4I have 2 children. The first one is F. (A condition is put on it.) What is P(2F)? We haveeliminated two possible outcomes, MF and MM.So the Punnett square is:P(FF|F1) = P(FF∩F1)/P(F1) = (1/4) /(1/2) = 1/2Punnett squares are ways of getting all possible combinations of things.e.g. all combinations of gametesIn cases we have considered, the probabilities are all equal. But they don’t have to be.Consider tossing a weighted coin which has probability of heads = 0.6 and tails 0.4.Toss it twice (or toss two such coins):What are the probabilities?P(HH) = (0.6)(0.6) = 0.36P(HT) = P(TH) = (0.6)(0.4) = 0.24P(TT) = (0.4)(0.4) = 0.16Check: 0.36 + 2(0.24) + 0.16 = 1We could use a Punnett Square as follows:Punnett Squares With Unequal Probabilities0.16 T T0.24 H T0.24 H T0.36 H H First toss 0.6 H 0.4 T SecondToss0.6 H0.4 TBinomial Probability DistributionProblem: Mendel observed approximately 3:1 ratio in the F2 of