A Little Intro to Statistics What’s the chance of rolling a 6 on a dice? 1/6 What’s the chance of rolling a 3 on a dice? 1/6 Rolling 11 times and not getting a 6? (5/6) 11 ~ 13.4% Rolling 11 times and not getting a 3? (5/6) 11 ~ 13.4% In a pile of Organic Chemistry Exams, what’s the chance of seeing a score between 75 and 78%? What about between 0 and 3%? What’s the chance of cancer cells surviving after being exposed to radiation for a period of time? –Period of time is short. –Period of time is long.

Examples of Different Types of Probability Distribution Exponential (i.e. survivability) Uniform (i.e. dice role) Normal (i.e. exam scores)

Sampling from a Distribution Take exam scores that are normally distributed Take a pile of exams Randomly pick one Record the score Do this enough times, and you’ll see a normal distribution  Can simulate taking samples with any known distribution. 

Recombination 2 Locus Example gamete Locus b Locus a Recombination location

2-Locus Recombination Example 1.Determine amount of time (backwards, in generations) when an event occurs. Locus a Locus b gamete 1gamete 2 t1t1 t 1  sampled from exponential distribution w/ mean 2N/(1+R) where R=4Nr, N is effective population size and r per generation per offspring is recombination rate. This comes from doing the statistics of taking into account that two events occur: coalescence and recombination. present past

2-Locus Recombination Example 2. Determine what type of event occurred Here, the events possible are coalescence or recombination. Locus a Locus b gamete 1gamete 2 t1t1 P coalesc = 1/(2N) P recomb = 2r Let’s consider that a recombination event occurs on gamete R R. 1 + R where R=4Nr, N is effective population size and r per generation per offspring is recombination rate. present past

2-Locus Recombination Example 3. Repeat... Determine t n, then determine type of event. Locus a Locus b gamete 1gamete 2 t1t1 Say next a coalescence event occurs between gamete 1 and locus a of gamete 2. P coalesc = 3 choose 2 /(2N) P recomb = r t 2  sampled from exponential distribution t2t2 CA a present past

2-Locus Recombination Example Repeat until you find MRCA (most recent common ancestor) for all samples at all loci. Locus a Locus b gamete 1gamete 2 t1t1 t2t2 CA a CA b present t3t3 past  Note that the genealogies for locus a is different than locus b 

Introduction of Mutations Locus a Locus b gamete 1gamete 2 t1t1 t2t2 CA a CA b x x xx present t3t3

Infinite site Model Now recombinations can occur at any location  Can use same procedure as the two loci, just use appropriate probabilities  Slightly different model: 1 kb region, 999 locations of recombination