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

Date:

July 22, 2026

Answering time:

60 minutes

Total points:

60

Name:

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Student ID:

UW NetID:

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Signature:

Instructions

Permitted materials

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

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. 10 points

A university student ID consists of three uppercase English letters followed by four digits. Letters and digits may repeat. The first letter cannot be X, and at least one of the four digits must be even.

How many such student IDs are possible? Justify your counting. _________________________________________

Problem 2. 10 points

A warehouse receives items from three factories. Factory \(A\) supplies \(40\%\) of the items and has a defect rate of \(2\%\). Factory \(B\) supplies \(35\%\) of the items and has a defect rate of \(5\%\). Factory \(C\) supplies \(25\%\) of the items and has a defect rate of \(8\%\).

An item is selected uniformly at random from the warehouse and is found to be defective.

(a)
Find the probability that the item came from Factory \(C\). (6 points)
(b)
Find the probability that the item did not come from Factory \(A\). (4 points)

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Problem 3. 10 points

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

(a)
Prove that \[ P(A\cup B\cup C) =P(A)+P(B)+P(C) -P(A\cap B)-P(A\cap C)-P(B\cap C) +P(A\cap B\cap C). \] (6 points)
(b)
Deduce that \[ P(A\cup B\cup C)\leq P(A)+P(B)+P(C). \] (2 points)
(c)
Give a simple necessary and sufficient condition for equality in part (b). (2 points)

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Problem 4. 15 points

The cumulative distribution function of a continuous random variable \(X\) is \[ F_X(x)= \begin {cases} 0, & x<0,\\[1mm] \dfrac {x^2}{4}, & 0\leq x<1,\\[2mm] \dfrac {2x-1}{4}, & 1\leq x<2,\\[2mm] 1-\dfrac {(3-x)^2}{4}, & 2\leq x<3,\\[2mm] 1, & x\geq 3. \end {cases} \]

(a)
Find the probability density function \(f_X(x)\). (4 points)
(b)
Prove that \(f_X(x)\) found in part (a) is a PDF. (3 points)
(c)
Find \(P\!\left (\frac 12<X\leq \frac 52\right )\). (3 points)
(d)
Find \(\mathbb {E}[X]\). (5 points)

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