STATS 2D03: Sample Final Exam
Drs. Noah Forman and J. E. Paguyo
December 2025
Question 1. A Kinder Joy chocolate egg either contains a toy or is empty. Suppose that each egg contains
a toy with probability 0.3, independently of other eggs. Moreover, suppose that each toy is colored red,
white, or blue with equal probability, independently of other toys. You buy 5 eggs. Let E1 be the event that
you get at most 2 toys and let E2 be the event that you get you get at least one toy of every color (i.e. at
least one red and at least one white and at least one blue toy).
(a) Compute P (E1 ).
(b) Compute P (E2 ). (Hint: Use the inclusion-exclusion principle.)
Question 2. You play the following game. You have an urn with 10 marbles: 2 red marbles, 3 blue marbles
labeled 1 to 3, and 5 green marbles. You draw one marble at random from the urn.
• If you draw a red marble, you win.
• If you draw a green marble, you lose.
• If you draw a blue marble, look at its label:
– If the number is 1, you lose.
– If the number is 2 or 3, you draw that number of additional marbles at random from the urn
(without replacement). If at least one of these additional marbles is red, you win. Otherwise, you
lose.
(a) What is the probability of winning this game?
(b) You played this game and won. What is the conditional probability that you drew the blue ball labeled
with a 3?
Question 3. You are a scientist studying dinosaurs. Every year, you collect 4000 eggs of a certain specimen
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probability of hatching, independently of other eggs.
of dinosaur. Each egg has a 1000
For these problems, you must use either the normal or Poisson approximation, with justification. For the
normal approximation, you are not required to use the continuity correction (though it is allowed).
(a) A year is called successful if at least 4 eggs hatch. Approximate the probability that a year is successful.
(b) Suppose you do this for 50 years. Approximate the probability that at least 30 years are successful.
Question 4. Let Z ∼ N (0, 1) and Y = eZ .
(a) Use Markov’s inequality to get an upper bound on P (Y > 10). That is, find a constant c so that
P (Y > 10) ≤ c.
(b) Use Chebyshev’s inequality to get an upper bound on P (Y > 10).
(Hint: You can find the expected value and variance of Y by thinking about the definition of the MGF and
using the MGF of the normal distribution.)
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Question 5. An urn has 2 red, 5 white, and 3 blue balls. Draw 3 balls at random (without replacement)
from the urn. Let X be the number of red balls and Y the number of white balls.
(a) Find the joint PMF of X and Y .
(b) Compute P (Y = 1).
(c) Compute E[XY ].
Question 6. Let X and Y be jointly continuous random variables with joint PDF
(
21 2
x y if x2 ≤ y ≤ 1,
f (x, y) = 4
0
otherwise
(a) Compute P (X ≥ Y ).
(b) Compute the marginal PDFs of X and Y .
(c) Are X and Y independent? You must justify your answer.
Question 7. Let X and Y be independent random variables such that X ∼ Binomial(10, 0.5) and Y ∼
Binomial(15, 0.5). Let V = X + Y .
(a) Compute P (V = 20).
(b) Find the MGF of V . (You may use the following fact without proof: the MGF of a Binomial(n, p)
random variable is M (t) = (1 − p + pet )n .)
(c) Compute E[V 2 ].
Question 8. Dr. Paguyo has office hours from 1:00pm to 3:00pm. Students arrive at his office according
to a Poisson process, with rate λ = 8 students per hour.
(a) What is the probability that two students arrive between 1:00pm and 1:30pm, no students arrive
between 1:30pm and 2:30pm, and three students arrive between 2:30pm and 3:00pm?
(b) What is the probability that the first student arrives before 1:30pm?
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Z-score table
Φ(−x) = 1 − Φ(x).
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Stats 2D03 final exam formula sheet
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