1. 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.

2. Examples of Different Types of Probability Distribution http://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm Exponential (i.e. survivability) Uniform (i.e. dice role) Normal (i.e. exam scores)

3. 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. 

4. Recombination 2 Locus Example gamete Locus b Locus a Recombination location

5. 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

6. 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 2. 1. 1 + R R. 1 + R where R=4Nr, N is effective population size and r per generation per offspring is recombination rate. present past

7. 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

8. 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 

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

10. 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