Little probability quiz
I teach combinatorics as a one-term elective to high school seniors (and a few juniors). We use Ivan Niven’s “Mathematics of Choice – How to Count Without Counting” for two marking periods. But we’ve reached a turning point. Two-thirds of the way through, we drop counting and turn our attention to probability, then games and expected value. We will finish with some analysis of rou- and I break the word up -ll- into three pieces -ette.
Since I have had nothing to post, but since I wrote a nice little probability worksheet, I thought I’d post that. To see the questions, click —->
Basic Probability
1. If there are three possible results, A, B and C, and P(A) = 0.3 and the P(B) = 0.4, find the P(C)
2a,b. Find P(x) + P(not x). Explain
Coins – binomial
3. What is the probability of getting exactly 9 heads in 12 throws?
4. What is the probability of getting at least 10 heads in 12 throws?
Coins – analysis
5a- i. Find the probability of getting exactly 3 heads for each situation:
2 throws, 3 throws, 4 throws, 5 throws, 6 throws, 7 throws, 8 throws, 9 throws, 10 throws.
5j. Organize your data on a graph or in a table.
5k. Does this match what most people would expect? Explain.
6a,b. What is the probability of getting exactly 20 heads in 40 throws. Does this contradict the idea of “evening out in the long run?”
Dice – basic
7a-f. Find each probability for one throw of one die:
P(5), P(>4), P(7), P(odd), P(2 or 4), P(not 3)
Two dice
8a-e. Find each probability for two dice (sum, unless otherwise noted):
P(2), P(odd), P(doubles), P(neither is a 5), P(product is even)
One red die and one blue die are thrown.
9a. Find P(red > blue)
9b. Carefully explain why the answer is not 0.5.
10. Find the probability that in 10 rolls of one die, there will be no 3’s
JD2718 
the notation’s pretty sloppy.
i might be prepared to forgive you
for trying to use “P(E)” notation
without defining the proper random variables–
students will fight to avoid learning definitions
with a ferocity math-heads reserve for dentistry
and suchlike — but anyhow i’m not about to
pass by an opportunity to try to talk you into
doing it right.
P(>4) is just flat-out *ugly*.
P(red>blue) might even be worse
(though it’s better *technically*;
one can *define* “red” as “the number on
the first die” [it’s just not a good *idea*]…
whereas “(>” will remain nonsense
*whatever* definitions we make
unless we throw out our context altogether).
when we get all slangy like this, it just makes it harder
to persuade the students that precise meanings
are even *possible* — which sounds like it oughta
be easy … but we both know it ain’t. most people
apparently believe that natural language
is as good as it gets and that insisting on code
that actually compiles is mere pedantry.
(even if they think good code *is* possible,
they sure as bejabbers don’t think it’s *useful*.)
we’re gonna have to keep fighting this battle
for the rest of our working lives. giving up
makes it harder on the next guy.
also a few more periods would be nice.
but my biggest gripe: guessing what
“most people would expect”
is family feud, not math.
oh, and “the P(C)” is pronounced
“the the probability of C”
(and isn’t good english).
why do you break up roullette?
Thanks for the comments.
First group – laziness. Not an excuse, and thanks for calling me on it.
“The the P of C” oops! Connecticut River. VIN Number… I should know better.
But the family feud comment? I think you need to be here. The ‘why people tend to get probability wrong’ aspect has been quite productive, because it makes them to grapple with what they thought before doing the math, and what they have figured out. Further, as they explain a misconception, they are forced to contrast it with good work. Does that make any sense to you?
.oh, and. I don’t want. search engines. bringing people. here for that game, that’s why.
(making up for some missing p.e.r.i.o.d.s)
Jonathan, the image you’ve linked to is causing my browser much sadness (I get some message about the MAA’s certificate being out of date).
You mentioned question 6 or variations before (last year when you were teaching this class?) and I still think it’s a nice one to make the solver really think about what’s going on.
Image deleted.
We spent more time this year creating discrete graphs, and watching how the center “clusters” as n gets large. I made them run enough calculations for enough n’s that most should have some sense in terms of numbers of what is going on.
Every year I teach, I get better. Things change. I change exercises, areas of investigation. But stuff that works well, I keep. And sometimes I might repeat some of that on here, as well. I hope that’s not a nuisance.
I don’t think it’s a problem at all if stuff repeats. It may remind us of something cool that we wanted to use and forgot about. And it may bring something cool to the attention of a new reader.
I’m all for repeating for the reasons mathmom listed!
I wasn’t complaining at all (except about the image — thanks for removing it!), I just don’t know if I’d remarked on the niceness of that problem last time around :)
Oh, yeah, I’ll add that my browser didn’t like the image either, so I also thank you for taking it down. :)