ECON3209 Assignment 1
Rachael Meager
Released: End of Week 2
DUE DATE: End of Week 3
The total amount of points on this assignment is 60. It includes a quiz section and a
long problem section. The entire assignment is worth 20% of your grade.
Answer all questions in all sections. This is open book and we highly encourage you
to use R to solve problems where possible. If you use R then you should copy and paste
your R code into your solution as your "working". When you are asked to make plots you
may do it by hand or in R, either is acceptable, unless otherwise specified in the question.
You are encouraged to work on this assignment with your classmates and discuss your
answers together, but you must write up (or draw up) your own answers separately.
Upload your solutions in 1 file to Moodle by 6pm on Friday of Week 3.
1 Quiz Questions
(18 points total) Each quiz question is worth 2 points.
1. I pick 3 integers from 1 to 10 without replacement. How many possible outcomes?
2. I pick 3 integers from 1 to 100 without replacement. How many possible outcomes?
3. I roll a standard dice 10 times. How many possible outcomes?
4. If the probabilities of two disjoint events A1 and A2 in a state space S are {0.2, 0.15}
respectively, what is Pr(S\{A1, A2}) ?
5. Suppose that you draw a pokemon at random (equiprobably) from the original 151
pokemon. The result is hidden. However, you are told that you have drawn an
electric type pokemon, of which there are 9. Given that you have drawn electric
type, what is the chance you drew pikachu?
6. Does an event A with Pr(A) = 0 necessarily have to be the empty set?
7. Is it true that for any set of n events E1, E2, ..., En,
P (∪ni=1Ei) =
n∑
i=1
P (Ei)?
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8. Can a uniform distribution be assigned to all the elements of Z (the integers)?
9. Say we have some PDF f(x) as follows:
f(x) =
{
cx if 0 < x < 10
0 otherwise
Find c.
2 Long Questions
(42 points total) Points awarded are indicated in brackets.
1. (7 pts) Clefairy is a fairy type pokemon, Zubat is a poison and flying type Pokemon,
Bulbasaur is a grass and poison type Pokemon, Scream Tail is a fairy and psychic
type Pokemon, Ponyta is a fire type Pokemon, Togetic is a fairy and flying type
Pokemon, and Charizard is a flying and fire type Pokemon. Answer the question
referring to this list.
(a) What is the set of fire type Pokemon? (1pt)
(b) What is in the union of fairy and grass type Pokemon? (1pt)
(c) What is the intersection of grass and flying type Pokemon? (1pt)
(d) What is in the intersection of grass and poison type Pokemon? What is the
union? (1pt)
(e) Draw a picture that shows the sets of fairy and flying type Pokemon, indicating
their union and intersection. (You can indicate each Pokemon by their names
if you dont want to try drawing them). (3 pts)
2. (5pts) Consider any two sets A and B.
(a) Prove that any element x ∈ (A ∩B)c is necessarily in Ac ∪Bc. (2 pts)
(b) Does this constitute a proof of DeMorgan’s Second Law? Explain why or why
not. (3 pts)
3. (6pts) You have an urn with 20 white and 7 red balls. You sequentially draw 3 balls
without replacement. A is the event that the first ball is white, B that the second
ball is white, C is the event that the third is white.
(a) What is the probability that the first ball is white? (1 pt)
(b) What is the probability that the second ball is white? (2pt)
(c) Conditional on the first draw being white, what is the probability that the
third ball is white? (3pts)
4. (2 pts) I am organising my pokemon figurine collection. I have 15 pokemon figurines.
I want to keep 7 in my office, 5 in my apartment, and 3 in my garden. How many
different possible ways can I do this?
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5. ( 4 pts) Consider an 8 sided dice, or "D8" according to our Dungeons and Dragons
friends. The event space of possible outcomes for this dice is {1, 2, 3, 4, 5, 6, 7, 8}.
A fair 8-sided dice rolls each number with probability 18 . However, suppose there
was an accident at the dice factory. This dice never rolls a 3. Instead, it is twice as
likely to roll a 4.
(a) Draw the PMF for this wonky D8, and make sure to label the y axis values
taken by the function. (2pt)
(b) Draw the CDF of this wonky D8, and make sure to label the y axis values
taken by the function. (2pt)
6. (3 pts) Use the binomial PMF to work out the probability that 15 flips of a fair coin
returns 5 heads.
7. (4 pts) Draw the CDF of the continuous uniform distribution over some interval
[a, b] on the reals. Be sure to show the value of the functions outside of this interval
also. What is the slope of this function in the interval? What about outside the
interval?
8. (5 pts) Stanford neuroscientist Andew Huberman, better known for his gig as a
podcast host, said on his podcast the following statement: "For women 30 years
old or younger, because the probability of getting pregnant on any one attempt to
conceive is 20%, well then if that doesn’t occur the first time then she should simply
try to repeat that at least 5 and probably 6 times... because 20 times 5 is 100, so
we’re talking about cumulative percent, so 20, 40, 60, 80, 100... and the 6th month
there would take you to 120%, which is a different thing altogether..."
What is wrong with this statement? Set up and do the correct calculation for the
probability of getting pregnant from 6 attempts. Take 20% chance on each attempt
as given, assume independence between attempts and assume that one does all 6
attempts before checking the result, to simplify.
9. (6pts) Suppose you are on a game show, and given the choice of 4 doors: Behind
one door is the desired prize, a car; behind the others, goats. (For this question you
are supposed to want a car and not want a goat.) You pick a door, say No. 1, and
the host, Monty, opens two other doors, say No. 3 and No. 4, which both have a
goat. He then says to you, "Do you want to pick door No. 2?" Assume that Monty
cannot open your door, D1, that he must open two of the remaining doors D2 or
D3 or D4, and he only ever opens doors with goats.
(a) Work out the probability door 2 has the car, given all the above.(4pts)
(b) Work out the probability door 1 has the car, given all the above. (1pt)
(c) Are the odds for switching more or less favourable now relative to the "3 door"
version of the problem we saw in lectures and tutorials? (1pt)
Tip: Wait until after Tutorial 2 to do this question.
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