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Final

Answers to chapters 2 and 3 for the second exam. (45%)

20 Pages
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Department
Philosophy
Course Code
PHL246H1
Professor
Colin Howson

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PHL246 answers provided by Charles Dalrymple-Fraser
Behold, an introduction!
1) Exercises
In this packet I have written up my answers for the exercises from
chapter two exercise five, through chapter three exercise four (those are
sort of the boundaries of the test content, so to speak).
I must note, as I did last time around, that my answers might be wrong,
or I might not have been clear. If you find such errors in here, email me
back at my utoronto.ca account and Ill try to fix them up so as to not
lead anyone astray with false answers. If you want some clarification,
you can also email me, and Ill do my darndest to get back to you (but if
I dont get around to it in time, please dont hate me: try the TA’s too;
Im just another student after all).
Finally, some people have emailed me asking if Id like compensation for
this. As modestly as I can project, Id like to say that I dont expect any
compensation for this, but if you want to buy me a coffee or whatever, I
certainly wont decline I have put time into making this as helpful as
possible.
2) Study Tips
A common question that has popped up in class during demonstrations is
some phrasing of but how did you get that? or how would we come up
with that?. Obviously, it is difficult sometimes to see where to start with
a question. So before the exercise answers, I have put together a very (I
emphasize the very) brief list of suggestions for your studying to help
you get right at the questions on the test and save some of that oh-so-
precious time. Of course, these are things I tend to do, and not everyone
thinks the same, these are just suggestions.
Best of luck in your studies and on Monday, Ill see you there.
Charles.
www.notesolution.com
PHL246 answers provided by Charles Dalrymple-Fraser
Some Study Tips (in no particular order)
1. Know those definitions. You can expect that you might have a question saying that A and B
are independent, and to prove something something. It will be a lot easier to answer the
question, chances are, if you recall what it means for, say A1,,An to be independent (i.e. if you
remember the equations P(±A1^±A2...^±An) = P(±A1)P(±A2) ... P(±An) and P(A|B)+P(A) youll
probably be better off). I would especially recommend looking terms from Chapter 3 as I expect
theyll arise on this test. Make sure you know at least (I might have missed some too):
independence, correlation, odds, bettering quotient (p), personal probability, fair odds, and fair
bettering quotients. They may or may not arise, but if they do, you dont want to be bumbling
about trying to recall what exactly the bettering quotients equation or symbol is.
2. Work to simplify, not expand. If you are asked to prove that, for example, P(A) = P(A|B)P(B)
+ P(A|~B)P(~B), you should probably work with the right side of the equation, trying to shrink it
down until you have it equal P(A). If you try to work up from P(A), its more likely that you will
get lost or use the wrong equation and waste precious time. I suggest working with the ugliest
thing whenever possible. For example, we solve this equation in exercise 5iii by simplifying the
right side using rule five, and then breaking those terms down even further until we get plain old
P(A). If you tried working from P(A), youll probably need a really firm idea of what you are
doing and where you are going before you start the question. When in doubt, try to simplify and
tackle the ugly things. I really want to stress this approach it should make things way easier
for you down the line.
3. Know your equations. It makes it a heck of a lot easier to try to do math when you actually
have equations to use. Do you remember the fourth probability rule right now? What about
Bayes Theorem and its more useful corollaries? What about the Betting Quotient for or against
A? Maybe its worth scanning through the chapter again and picking out some equations.
4. Know your terms. If you know your equations, you want to know when to use them. It might
seem simple, but if you can pick up right away that P(A^B) is connected to rule five, or that if A
and B are mutually exclusive, you can use rule four, or that given the odds and total stakes you
can calculate individual stakes, then you will be saving a lot of time trying to figure out where to
go next. If you dont know for example, how to get rid of a term like P(B|A)P(A) right away (rule
five or Bayes Theorems), you might need to practice with figuring out what terms fit in what
equations.
5. Dont just try the exercises: try the examples too. Last test, we had to prove at least one of
the examples from the chapter (I dont recall if more); its likely that youll have to again. Look
over the examples and the equations from the chapter. Try working them out, proving them.
Where the text doesnt blatantly state the proof, it usually tells you what you need to do it.
6. Start with what you know. There are many approaches to test taking. For these tests, I would
suggest that you definitely start with what you know, regardless of the weight distribution of
questions. This might seem like a no-brainer, but really consider it. Theyll get you your easy
points and leave you a set amount of time to do the harder ones, rather than be out of time for
the easier ones. Plus, youll have a better short-term foundation for problem solving, and be
more likely to get into the swing of things and creative approaches.
7. Study Patterns. What has come up a lot is no doubt important and more likely to show up
again later. Think about what percentages of questions appeal to rule five of probability.
Prioritize your learning/studying by what is most prominent almost as equally as what you dont
know as well.
Thats all I got for now. Take it or leave it. Here are the exercise answers:
www.notesolution.com
PHL246 answers provided by Charles Dalrymple-Fraser
Chapter 2, Exercise 5.
Show that if P(B) WKHQ
i. P(T|B) = 1
ii. If A and C are mutually exclusive, P(AvC|B) = P(A|B) + P(C|B).
iii. P(A) = P(A|B)P(B) + P(A|~B)P(~B).
iv. If B => A then P(A|B) = 1. (Hint: Use (3) in Chapter 1).
I. P(T|B)=1
P(T|B) = P(T^B) from rule five
P(B)
T^B Ù B from chapter one; draw a truth table if needs be
P(T^B) = P(B) from rule one
P(T|B) = P(B) from substitution
P(B)
P(T|B) = 1 QED
II. P(AvC|B) = P(A|B) + P(C|B)
To start, lets expand the left side of the equation. Then, we will try to get the right
side to equal it.
P(AvC|B) = P((AvC)^B) from rule five
P(B)
P(AvC|B) = P((A^B)v(B^C)) from distributive law
P(B)
P(AvC|B) = P(A^B)+P(B^C) from rule four, being exclusive
P(B)
P(AvC|B) = P(B)P(A|B)+P(B)(C|B) from rule five
P(B)
P(AvC|B) = P(B)[P(A|B)+(C|B)] simplifying
P(B)
P(AvC|B) = P(A|B)+(C|B) after dividing out P(B)
QED
www.notesolution.com

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Description
PHL246 answers provided by Charles Dalrymple-Fraser Behold, an introduction! 1) Exercises In this packet I have written up my answers for the exercises from chapter two exercise five, through chapter three exercise four (those are sort of the boundaries of the test content, so to speak). I must note, as I did last time around, that my answers might be wrong, or I might not have been clear. If you find such errors in here, email me back at my utoronto.ca account and Ill try to fix them up so as to not lead anyone astray with false answers. If you want some clarification, you can also email me, and Ill do my darndest to get back to you (but if I dont get around to it in time, please dont hate me: try the TAs too; Im just another student after all). Finally, some people have emailed me asking if Id like compensation for this. As modestly as I can project, Id like to say that I dont expect any compensation for this, but if you want to buy me a coffee or whatever, I certainly wont decline I have put time into making this as helpful as possible. 2) Study Tips A common question that has popped up in class during demonstrations is some phrasing of but how did you get that? or how would we come up with that?. Obviously, it is difficult sometimes to see where to start with a question. So before the exercise answers, I have put together a very (I emphasize the very) brief list of suggestions for your studying to help you get right at the questions on the test and save some of that oh-so- precious time. Of course, these are things I tend to do, and not everyone thinks the same, these are just suggestions. Best of luck in your studies and on Monday, Ill see you there. Charles. www.notesolution.comPHL246 answers provided by Charles Dalrymple-Fraser Some Study Tips (in no particular order) 1. Know those definitions . You can expect that you might have a question saying that A and B are independent, and to prove something something. It will be a lot easier to answer the question, chances are, if you recall what it means for, say A ,1A to ne independent (i.e. if you remember the equations P(A ^A ...^A ) = P(A )P(A ) ... P(A ) and P(AB)+P(A) youll 1 2 n 1 2 n probably be better off). I would especially recommend looking terms from Chapter 3 as I expect theyll arise on this test. Make sure you know at least (I might have missed some too): independence, correlation, odds, bettering quotient (p), personal probability, fair odds, and fair bettering quotients. They may or may not arise, but if they do, you dont want to be bumbling about trying to recall what exactly the bettering quotients equation or symbol is. 2. Work to simplify, not expand. If you are asked to prove that, for example, P(A) = P(AB)P(B) + P(A~B)P(~B), you should probably work with the right side of the equation, trying to shrink it down until you have it equal P(A). If you try to work up from P(A), its more likely that you will get lost or use the wrong equation and waste precious time. I suggest working with the ugliest thing whenever possible. For example, we solve this equation in exercise 5iii by simplifying the right side using rule five, and then breaking those terms down even further until we get plain old P(A). If you tried working from P(A), youll probably need a really firm idea of what you are doing and where you are going before you start the question. When in doubt, try to simplify and tackle the ugly things. I really want to stress this approach it should make things way easier for you down the line. 3. Know your equations. It makes it a heck of a lot easier to try to do math when you actually have equations to use. Do you remember the fourth probability rule right now? What about Bayes Theorem and its more useful corollaries? What about the Betting Quotient for or against A? Maybe its worth scanning through the chapter again and picking out some equations. 4. Know your terms. If you know your equations, you want to know when to use them. It might seem simple, but if you can pick up right away that P(A^B) is connected to rule five, or that if A and B are mutually exclusive, you can use rule four, or that given the odds and total stakes you can calculate individual stakes, then you will be saving a lot of time trying to figure out where to go next. If you dont know for example, how to get rid of a term like P(BA)P(A) right away (rule five or Bayes Theorems), you might need to practice with figuring out what terms fit in what equations. 5. Dont just try the exercises: try the examples too. Last test, we had to prove at least one of the examples from the chapter (I dont recall if more); its likely that youll have to again. Look over the examples and the equations from the chapter. Try working them out, proving them. Where the text doesnt blatantly state the proof, it usually tells you what you need to do it. 6. Start with what you know. There are many approaches to test taking. For these tests, I would suggest that you definitely start with what you know, regardless of the weight distribution of questions. This might seem like a no-brainer, but really consider it. Theyll get you your easy points and leave you a set amount of time to do the harder ones, rather than be out of time for the easier ones. Plus, youll have a better short-term foundation for problem solving, and be more likely to get into the swing of things and creative approaches. 7. Study Patterns. What has come up a lot is no doubt important and more likely to show up again later. Think about what percentages of questions appeal to rule five of probability. Prioritize your learningstudying by what is most prominent almost as equally as what you dont know as well. Thats all I got for now. Take it or leave it. Here are the exercise answers: www.notesolution.com
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