Lectures 5 and 6: Introduction to Probability theory

(version 7 Jan 2008)

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The birthday problem:

Introduction to Probability

Useful Rules of Probability

Example: ABO blood groups

Recall that there are four possible phenotypes for the ABO blood group: Type A, Type AB, Type B, and Type O.

Suppose Prob(Type A) = 0.2, Prob(Type B) = 0.3, Prob(Type) AB= 0.1. What is Prob(Type O)?

Likewise, what is the probabilty that you are NOT type O?

The AND and OR Rules

Example: Expected Marker Genotype Frequencies

Suppose you have a forensic DNA maker that has five alleles with the following frequecies:

Freq(A1) = 0.1, Freq(A2) = 0.2, Freq(A3) = 0.2, Freq(A4) = 0.4, Freq(A5) = 0.1,

What the expected frequency of an A5A5 homozygote?

To get an A5A5, that individual must get an A5 from its father and get an A5 from its mother.

If we have random mating, then

Pr(A5 from father AND A5 from mother) = Pr(A5)*Pr( A5 from mother) = 0.12 = 0.01

What the expected frequency of an A1A2 heterozygote?

There are two ways for this to happen: A1 from father and A2 from mother or A2 from father and A1 from mother.

Pr(A1 from father and A2 from mother or A2 from father and A1 from mother)

= Pr(A1 from father and A2 from mother) + Pr( A2 from father and A1 from mother).

= Pr(A1 from father)*(Pr A2 from mother) + Pr( A2 from father)*Pr( A1 from mother).

= 0.1*0.2 + 0.2*0.2 = 2*0.1*0.2 = 0.04

Conditional Probability

How do we compute joint probabilities when A and B are NOT independent (i.e., knowing that A has occurred provides information on whether or not B has occurred).

Example

Example: Lotto

Consider the Arizona State Lottery, wherein you pick 6 numbered balls out of 40. If all six of your balls are drawn, you win. What is the chance of this happening?

Prob(win jackpot) = (6/40)*(5/39)*(4/38)*(3/37)*(2/36)*(1/35) = 1/ 5,245,786


How long must one play lotto to have a reasonable (say 50 percent) chance of winning the jackpot?

Suppose you buy 100 different lotto tickets for each drawing. How many such drawings do you have to play to have a 50 percent chance of winning (at least) one jackpot?

Probabilities for the birthday problem