Department

Philosophy

Course Code

PHL246H1

Professor

Colin Howson

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 I’ll 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 I’ll do my darndest to get back to you (but if

I don’t get around to it in time, please don’t hate me: try the TA’s too;

I’m just another student after all).

Finally, some people have emailed me asking if I’d like compensation for

this. As modestly as I can project, I’d like to say that I don’t expect any

compensation for this, but if you want to buy me a coffee or whatever, I

certainly won’t 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, I’ll 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) you’ll

probably be better off). I would especially recommend looking terms from Chapter 3 as I expect

they’ll 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 don’t want to be bumbling

about trying to recall what exactly the bettering quotient’s 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), it’s 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), you’ll 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 it’s 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 don’t 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. Don’t just try the exercises: try the examples too. Last test, we had to prove at least one of

the examples from the chapter (I don’t recall if more); it’s likely that you’ll have to again. Look

over the examples and the equations from the chapter. Try working them out, proving them.

Where the text doesn’t 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. They’ll 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, you’ll 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 don’t

know as well.

That’s 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, let’s 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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