University of Washington
Department of Mathematics Midterm Examination
MATH 394: Probability I
Summer 2026 — Instructor: Arman Jahangiri

Date:	July 22, 2026
Answering time:	60 minutes
Total points:	60

Name:	______________________________________________________
Student ID:
Signature:	______________________________________________________

Instructions

This examination contains five problems. Verify that your copy is complete.
Show all work and justify your answers. Unsupported answers may receive little or no credit.
Clearly define any events or random variables that you introduce.
Do not remove pages from the examination booklet.

Permitted materials

Examinations are closed-book and closed-notes. Students may use:

one handwritten 8.5 \(\times \) 11 inch reference sheet (front and back), and
a non-programmable non-graphing calculator capable of standard exponential and logarithmic computations (e.g. TI-30X IIS).

No other electronic devices, calculators, communication devices, or external resources may be used during examinations.


Problem	Score

1	/10

2	/10

3	/10

4	/15

5	/15

Total	/60

By signing above, you affirm that the work is your own and that you followed the examination rules.

Problem 1. 5 points

A student organization has 7 mathematics graduate students, 5 statistics graduate students, and 6 undergraduate students. A committee of 5 students is selected uniformly at random from all committees of 5.

Find the probability that the committee contains exactly 2 mathematics graduate students, exactly 1 statistics graduate student, and exactly 2 undergraduate students.________________________________________________

Problem 2. 10 points

A spam filter is designed by looking at commonly occurring phrases in spam. Suppose that \(80\%\) of email is spam. In \(10\%\) of the spam emails, the phrase “free money” is used, whereas this phrase is only used in \(1\%\) of non-spam emails. A new email has just arrived, which does mention “free money.” What is the probability that it is spam?______________________________________________________________________________________________

Problem 3. 10 points

Let \(A\) and \(B\) be events.

(a): Let \(S\) be a sample space and \(A_1,\dots , A_n\in \mathcal {F}\) be events. Explain what it means for \(A_i's\) to form a partition of \(S\), (3 points)
(b): Suppose \(P(A) = 0.6\) and \(P(B)=0.5\). Can \(A\) and \(B\) be disjoint? (3 points)
(c): Suppose that \(A\) is independent of itself. Use the definition of independence to conclude that \(P(A)=0\) or \(P(A)=1\). (4 points)

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Problem 4. 15 points

Suppose that \(X\) has probability density function

\[ f_X(x)= \begin {cases} c\,x(2-x), & 0<x<2,\\ 0, & \text {otherwise}, \end {cases} \]

where \(c\) is a constant.

(a): Find \(c\). (3 points)
(b): Find the cumulative distribution function \(F_X(x)\) for all real \(x\). (5 points)
(c): Find \(P(X>3/2)\). (2 points)
(d): Find \(\mathbb {E}[X]\) and \(\operatorname {Var}(X)\). (5 points)

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Problem 5. 20 Points

A company has found that \(40\%\) of customers who visit its website make a purchase. Suppose that \(200\) customers visit the website independently on a particular day.

Let \(X\) denote the number of customers who make a purchase.

1.

Determine whether a Poisson approximation or a normal approximation is more appropriate by verifying the required conditions. Then clearly state the approximating distribution; that is, provide \(\lambda \) if a Poisson approximation is appropriate, or provide \(\mu \) and \(\sigma ^2\) if a normal approximation is appropriate. (5 points)

2.

Use the approximation from part (a) to estimate each of the following:

(a): \(\mathbb {P}(70<X<90)\) (5 points)
(b): \(\mathbb {P}(X>90)\) (5 points)
(c): \(\mathbb {P}(X=80)\) (5 points)

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