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University of Toronto Scarborough
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Statistics
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STAB22H3
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Ken Butler
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Lecture 10

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Statistics

STAB22H3

Ken Butler

Fall

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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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