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Statistics (281)
STAB22H3 (223)
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Lecture 10

# STAB22-LEC10-(11)-MID.docx

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Department
Statistics
Course
STAB22H3
Professor
Ken Butler
Semester
Fall

Description
STAB22 LEC10 NOTE: This lecture ONLY lasted till 34:12; it finishes up Chapter 11 Some notes about changes to his webpage (see the page here: http://utsc.utoronto.ca/~butler/b22 - About EXAM - from C1 to C11 - CONTENT EXEMPTED FROM EXAM? - timeplots - Plan B: Attack of the Logarithsm - do not have to calculate a correlation by hand TA OFFICE HOURS - IC404 (check the timings for Fri, Mon) WHEN IS THE EXAM? - Exam at Mon; 5-7pm ----------------------------- CHAPTER 11 - UNDERSTANDING RANDOMNESS [128] - The following games are random - tossing a coin - rolling a 6-sided dice - For sth to be random, it is - unexpected (unpredictable) in the short-term, but is - predictable in the long-term (ie. if you do trials over and over again, what will your likelihood be of getting X?) (ex) that shows that randomizations are predictable in the long-term: a) coin: - should get heads or tails about 1/2 of the time b) die - should roll a certain number from 1 to 6 about 1/6 of the time (ex) take randomizing device (ex. dice); if they are rolled, do not know what exactly will come up in the next roll (ie. in the short-term), but will know what will likely turn up to be in the long-term Computer random numbers - they are gen'ed by a computer algorithm - aka pseudorandom numbers, b/c they are prod'ed by a non-random method => the computer can make the #'s look like as if they are random, even tho. they are entirely prod'ed by a nonrandom method [129] RANDOM DIGIT TABLES What is a simulation? - It is sth that models that real-world scenario by using random-digit outcomes to imitate the uncertainty of a response var. of interest - ie. they are copying the probability of a certain val. of the response var. to appear, but depicting so with random numbers - can do simulation easily via a computer (ie. get a statistics package to list a ton of random digits for you) - OR, can use a pre-made table of random digits ranging from 0-9 - it is equally likely for any particulra digit to come, but knowing what the next digit is is unpredictable. => each digit is unpredictable given the numbers before it, but in the long run, each digit appears about the same # of times Why are they grouped in 5s? - a convention that is followed to keep track of where the digits are [130] SIMULATION: DICE GAME OF 21 - Play with 6-sided dice - Each player continues to roll a die, totalling up the numbers he has rolled until they: a) decide to stop b) go over 21, and in this case, they lose, and the other player wins - Player with the highest score that is less than or equal to 21 is the winner (Ex) - Butler rolls the dice and continually does so until he chooses to stop - and he gets: - 4, 5, 4, 6, 3, 2 = 22 - lost - The other player automatically wins - randomizing device used was dice - Question: You are playing against an opponent who scored 18. What is the likelihood of you winning? - strategy that You (the 2nd player) should take on is continue rolling until you win or go beyond 21. => you will end up either winning, or go over the upper limit and thus lose Are we still using the dice now to predict this probability? - No; we will run a simulation for this 21 game for the following situation: if the opponent goes first and scores 18, what is the chances of the 2nd player winning? [131] USING RANDOM NUMBERS TABLE - what we are doing is choosing a set of numbers from the table at random and listing them out - Note: in the context of this game, you can only get values from 1 to 6. - So, all random digits that are 0, 7, 8 and 9 do not count - one trial is one set of digits, in this case => the number of times the player rolls has a maximum of 5, and a min. of 0. (ex) say you get 24513  all these 5 numbers are on the die, so from this set of random val's, we say that the player choose to roll 5 times (ex) say you get 99990 chose to roll 5 times and got 22, which is over 21 so we lose ex2. 67663 - 6 + 6 + 6 + 3 = 21 => chose to roll 5 times and got 21, which is over 18 so we won. ex. 62561 - 6 + 2 + 5 + 6 + 1 = 20 => chose to roll 5 times and got 20, which is over 18 so we won. => after 3 trials, can see we won in 2 out of 3 trials (not simulations), but have to do many more to really get a accurate depiction of the probability of winning (ex) - He used another statistical software package and run 1000 simulations: - found that 2nd player has about 72% chance of winning [132] HERE'S A BIGGER TABLE, ALL DONE BY SIMULATION - The table is showing: if the first player scores a given total, then what is the likelihood that the second player will win? - he deduced the percentages from running a 1000 trials
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