1. Expectations 1 Krishna.V.Palem Kenneth and Audrey Kennedy Professor of Computing Department of Computer Science, Rice University

2. Contents Discussion of tutorial solutions Expectations & In-class exercise Sum of Expectations In-class exercise Project Discussion 2

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4. Contents Discussion of tutorial solutions Expectations & In-class exercise Sum of Expectations In-class exercise Project Discussion 4

5. In-class exercise -1 5 Let us change gears and move to a new topic Consider the following exercise Divide yourselves into teams of 2 and each person (A & B) gets equal number of chips, say 30 Each group will be given a coin to toss If you get a head, person A gets 2 chips from person B If you get a tail, person B gets 2 chips from person A After 20 rounds, how many more chips do you think each of you will earn (i.e. number of extra chips than the given 30 chips)? Let us test it out for ourselves.

6. 6 Maintain the record of your game in the following way TossOutcome Number of chips each player has 1 2 3 4 5... 19 20 Observe these numbers Can you now take a guess of your earnings after 50,100 or 1000 turns? Play this game for 20 rounds

7. 7 Can you now take a guess of your earnings after 50 turns, 100 turns, 1000 turns ? Answering this question is very important to get an idea of how much you are going to earn. Let us use probability First let us calculate the earnings each player can expect in one turn ip HEAD=0 ½ TAIL=1 ½ So a player has ½ chance of earning 2 chipsHe has ½ chance of losing 2 chips So we can expect to earn (½ * 2)- (½ * 2) = 0 more chip at the end of every turn on an average Very important

8. 8 Let us see another example This time we will consider another randomizing device – our friendly die Earnings OutcomeEarning 13 23 32 42 51 61 ThrowOutcome Number of chips till that point 1 2 3 4 5 6 7 8 9 10 Maintain a similar record till 20 throws in the following format

9. 9 We again visit the same question. Can you now take a guess of your earnings after 20 turns, 50 turns, 100 turns ? This time we know that the probability function of a die can be represented as ip(i) 11/6 2 3 4 5 6 There is a 1/6 th chance of winning 3 chips There is a 1/6 th chance of winning 2 chips There is a 1/6 th chance of winning 2 chip There is a 1/6 th chance of winning 1 chip On an average we can expect (1/6)*3 + (1/6)*3 + (1/6)*2 + (1/6)*2+ (1/6)*1 + (1/6)*1 = 2 chips every throw What does this mean ?

10. 10 What do we mean when we say that the earning per game is 2 chips? To understand this, let us first consider the following question. Consider an unbiased coin toss. The probability of obtaining a HEAD = ½ But for n trials of the experiment do we always get n/2 HEADs and n/2 TAILs ? Consider the following experiment: Toss a coin 5, 10, 50, 100, 500, 1000 … 10000 times. At each point collect the data regarding number of HEADs and number of TAILs. Now let us analyze data obtained from one such experiment.

11. 11 What do you observe as the number of trials grows large ?

12. 12 Can you observe that as the number of trials grows “large” the result of the experiment tends to agree with the ideal case ?

13. 13 What do we mean when we say that the earning per game is 2 chips? That does not seem right!! To understand this, let us first consider the following question. Consider an unbiased coin toss. The probability of obtaining a HEAD = ½ But for n trials of the experiment do we always get n/2 HEADs and n/2 TAILs ? Consider the following experiment: Toss a coin 5, 10, 50, 100, 500 … 9999 times. At each point collect the data regarding number of HEADs and number of TAILs. It means that as the number of trials(n) grows “large” then it can be expected that the earnings will be equal to 2* n Conclusion

14. Expectation 14 Consider the following table Event Random variable (x) Earning e(x) Probability P(x) Event 1x=131/6 Event 2x=231/6 Event 3x=321/6 Event 4x=421/6 Event 5x=511/6 Event 6x=611/6 Expected earning OR Average earning per Event (E) It can be written according to the previous intuition as e(1)*P(1) + e(2)*P(2) + … e(6)*P(6) =(3+3+2+2+1+1)*1/6 = 2 We can also define the earnings for each roll (or event) as a function of the random variable. So let us calculate the

15. Expectation Precise mathematical definition of expectation For a given experiment, there is an event space that is defined. A random variable x is defined which takes a distinct value for each event in the event space. A function f can be defined on the event space and thus on the random variable x The expected value of f is defined as Where i is the number of events

16. In-class exercise 2: Expectation Using the definition of expectation, solve the following problem An American roulette wheel has 38 places where the ball may land, all equally likely. A winning bet on a single number pays 35-to-1, meaning that the original stake is not lost, and 35 times that amount is won, so you receive 36 times what you've bet. This means that if you bet $1 on a specific one of the 38 places (i)If you lose, you lose the amount you bet (ii)If you win, you win 35 times the bet amount plus you keep you keep the amount you bet Question: 1.What is the expected amount of money you will be left with after each game? (Assume $1 bets) A) -$0.0263

17. Contents Discussion of tutorial solutions Expectations & In-class exercise Sum of Expectations In-class exercise Project Discussion 17

18. Expectation of a sum 18 Till now we have been referring to the expectation of a single quantity Suppose we need to evaluate the expected value of the sum of two dice throws Consider the following exercise Each person will get a pair of dice, say die 1 and die 2 The first round Each player rolls each die separately 20 times and calculates the expected value of each individual dice The second round Each player rolls both the dice at once 20 times and calculates the expected value of the total Simplifying assumption: Use 1/6 th as the probability for each outcome of the die

19. Expectation of a sum 19 Let us first try to evaluate mathematically what the expected value of the sum might be. Recall the basic formula for expectation The expected value of f is defined as Where i is the number of events

20. Analysis of data 20 RollDie 1 outcomeDie 2 outcome 1 2.. Calculate the expected outcome of each die from the above data RollDie 1 + Die 2 1 2.. Calculate the expected value of the outcome of both the dice together Let us check if the mathematical derivation is in fact correct

21. Joint Probability 21 Recall that we studied about joint or composite events For example, a roll of a die is a single event But roll of two dice simultaneously is a joint or composite event consisting of two simple events The probability of such an event can be expressed a joint probability function as follows : Consider the roll of two dice where x : Random variable that denotes the outcome of the first die y : Random variable that denotes the outcome of the second die then Probability that x = i and y=j is denoted as p(x=i,y=j) In fact in this case p(x=i,y=j) = p(x=i) * p(y=j) Here p(x=i,y=j) is called the joint probability of the two random variables x and y

22. 22 Let us define the following variables x : Random variable that denotes the outcome of the first die y : Random variable that denotes the outcome of the second die z : Random variable that denotes the outcome of the sum of both the dice Therefore we can say that z = x+y Using the formula for calculating expectation of z Cover all possible combinations of x and y

23. 23 Joint probability in this case is the product of the two probabilities Rearranging some terms Substitute the formula for expectation with E(Y)

24. 24 Sum of probabilities is equal to 1 Expanding the product inside the summation Substitute formula for expectation of X Sum of probabilities is equal to 1 Is a very important result

25. Lessons Learnt 25 How can you relate the expectations of individual dice and the total ? The expectation of the total is the sum of the individual expectations How do you mathematically prove it ? Use the formula for expectation Because it is a linear summation, But what are the conditions when this statement holds ? This statement holds under all conditions That means that this law can be applied on any set of random variables The expectation of a sum is equal to the sum of expectation

26. Generalizing the result 26 Prove that the expectation of sum of n random variables is equal to the sum of expectation of the n random variables. That is prove the following Let X 1, X 2, X 3 …. X n be n random variables Let Z = X 1 + X 2 + X 3 …. + X n To prove:

27. Contents Discussion of tutorial solutions Expectations & In-class exercise Sum of Expectations In-class exercise Project Discussion 27

28. Field Goal Percentage vs. Free Throw Percentage A case study of applied probability on sports By Yung-Seok Kevin Choi It is advantageous to quantify different aspects of a game to strategize In basketball two of the primary statistics are Field Goal Percentage ratio of field goals achieved to field goals attempted Free Throw Percentage ratio of free throws achieved to free throws attempted Solution Collect the data from a basketball reference website MATLAB to analyze and parse data The data was organized into different bins MATLAB to compute conditional probabilities such as Made statements such as “given a guard’s free throw percentage in bin 6, we are 85 percent confident that his field goal percentage falls between bins 6 and 11.” Problem Statement: In basketball, is field goal percentage related to free throw percentage and how does this relationship differ between positions?

29. Impartiality in Mafia By Jose Luis Garcia There is an interactive social game called Mafia The basic idea of the game is that there are two groups (Townspeople and Mafia) with each having different advantages who attempt to use these strengths in order to incapacitate the rival group and thus win the game. Is it the case that one of the two groups has more chance of winning than the other Specific Analysis What is the probability of being assigned to either mafia or townspeople? How many turns on average does it take to finish a game? Is there a group favored to win and if so which group? How is the win/loss probability related to the number of players? Solution Mathematical model of the game Define the variables (number of players, win probability etc.) MATLAB script to run multiple trials of the game Based on the ratio of townspeople to the mafia Define the probability of winning of the townspeople A card is chosen at random from a given stack (MATLAB script) Compute winning and losing probabilities based on multiple trials One such Conclusion As the number of players is increased the probability of Mafia winning is also increased. Problem Statement: Analyze the impartiality of an interactive social game called Mafia

30. Probability in Currency Exchange By Thomas Roinesdal The values of international currencies fluctuate every day Data about trading between among different currencies is available It will be very advantageous if one can predict the future value of the exchange rate of a currency based on the historical trends. Solution Model Consider pairs of currencies Example: 1 US dollar = 1.4 Singapore dollar Collect historical data of trends in the exchange rate Compute the relative frequency of different exchange rates Calculate conditional probabilities based on this data by applying Bayes Theorem Conclusion Based on historical data of many currency exchange rates, there was less than 30% chance of predicting accurately a future value of a given currency But it was shown that some currencies are more correlated than others Problem Statement: Is it possible to predict a currency given the historical values of other currencies?

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