15 251 Some Great Theoretical Ideas in Computer Science for Probability Theory I Lecture 11 February 17 2009 Some Puzzles Teams A and B are equally good In any one game each is equally likely to win What is most likely length of a best of 7 series Flip coins until either 4 heads or 4 tails Is this more likely to take 6 or 7 flips 6 and 7 Are Equally Likely To reach either one after 5 games it must be 3 to 2 chance it ends 4 to 2 chance it doesn t Silver and Gold A bag has two silver coins another has two gold coins and the third has one of each One bag is selected at random One coin from it is selected at random It turns out to be gold What is the probability that the other coin is gold 3 choices of bag 2 ways to order bag contents 6 equally likely paths Given that we see a gold 2 3 of remaining paths have gold in them Sometimes probabilities can be counter intuitive Language of Probability The formal language of probability is a very important tool in describing and analyzing probability distribution Finite Probability Distribution A finite probability distribution D is a finite set S of elements where each element t in S has a non negative real weight proportion or probability p t The weights must satisfy t S p t 1 For convenience we will define D t p t S is often called the sample space and elements t in S are called samples Sample Space 0 1 0 17 0 13 0 11 0 2 0 0 13 0 1 S 0 06 Sample space weight or probability of t D t p t 0 2 Events Any set E S is called an event PrD E t E p t 0 17 0 0 13 0 1 PrD E 0 4 S Uniform Distribution If each element has equal probability the distribution is said to be uniform PrD E t E p t E S A fair coin is tossed 100 times in a row What is the probability that we get exactly half heads Using the Language The sample space S is the set of all outcomes H T 100 Each sequence in S is equally likely and hence has probability 1 S 1 2100 Visually S all sequences of 100 tosses t HHTTT TH p t 1 S Event E Set of sequences with 50 H s and 50 T s Set of all 2100 sequences H T 100 Probability of event E proportion of E in S 100 2100 50 Suppose we roll a white die and a black die What is the probability that sum is 7 or 11 Same Methodology S 1 1 2 1 3 1 4 1 5 1 6 1 1 2 2 2 3 2 4 2 5 2 6 2 1 3 2 3 3 3 4 3 5 3 6 3 1 4 2 4 3 4 4 4 5 4 6 4 1 5 2 5 3 5 4 5 5 5 6 5 1 6 2 6 3 6 4 6 5 6 6 6 Pr E E S proportion of E in S 8 36 23 people are in a room Suppose that all possible birthdays are equally likely What is the probability that two people will have the same birthday And The Same Methods Again Sample space W 1 2 3 366 23 t 17 42 363 1 224 177 23 numbers Event E t W two numbers in t are same What is E Count E instead E all sequences in S that have no repeated numbers E 366 365 344 W 36623 E W 0 494 E 0 506 W More Language Of Probability The probability of event A given event B is written Pr A B and is defined to be Pr A B Pr B S B proportion of A B A to B Suppose we roll a white die and black die What is the probability that the white is 1 given that the total is 7 event A white die 1 event B total 7 S 1 1 2 1 3 1 4 1 5 1 6 1 1 2 2 2 3 2 4 2 5 2 6 2 Pr A B 1 3 2 3 3 3 4 3 5 3 6 3 Pr A B Pr B event A white die 1 1 4 2 4 3 4 4 4 5 4 6 4 1 5 2 5 3 5 4 5 5 5 6 5 A B B 1 6 2 6 3 6 4 6 5 6 6 6 1 6 event B total 7 Independence A and B are independent events if Pr A B Pr A Pr A B Pr A Pr B Pr B A Pr B Independence A1 A2 Ak are independent events if knowing if some of them occurred does not change the probability of any of the others occurring E g A1 A2 A3 are independent events if Pr A1 A2 Pr A1 Pr A2 A1 Pr A2 Pr A3 A1 Pr A3 Pr A1 A2 A3 Pr A1 Pr A2 A1 A3 Pr A2 Pr A3 A1 A2 Pr A3 Pr A1 A3 Pr A1 Pr A2 A3 Pr A2 Pr A3 A2 Pr A3 Silver and Gold One bag has two silver coins another has two gold coins and the third has one of each One bag is selected at random One coin from it is selected at random It turns out to be gold What is the probability that the other coin is gold Let G1 be the event that the first coin is gold Pr G1 1 2 Let G2 be the event that the second coin is gold Pr G2 G1 Pr G1 and G2 Pr G1 1 3 1 2 2 3 Note G1 and G2 are not independent Monty Hall Problem Announcer hides prize behind one of 3 doors at random You select some door Announcer opens one of others with no prize You can decide to keep or switch What to do Monty Hall Problem Sample space prize behind door 1 prize behind door 2 prize behind door 3 Each has probability 1 3 Staying we win if we choose the correct door Switching we win if we choose the incorrect door Pr choosing correct door 1 3 Pr choosing incorrect door 2 3 Why Was This Tricky We are inclined to think After one door is opened others are equally likely But his action is not independent of yours Next we will learn about a formidable tool in probability that will allow us to solve problems that seem really really messy If I randomly put 100 letters into 100 addressed envelopes on average how many letters will end up in their correct envelopes On average in class of size m how many pairs of people will have the same birthday The new tool is called Linearity of Expectation Random Variable To use this new tool …
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