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
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: