Practice Discrete vs Continuous in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The distinction between quantities that take separate, distinct values (discrete, like number of students) and quantities that can take any value in a range (continuous, like height or temperature).

People come in whole numbers (discrete). Height can be any value (continuous).

Showing a random 20 of 50 problems.

Example 1

hard
Explain why we use a probability density function (PDF) rather than a probability mass function (PMF) for continuous variables.

Example 2

medium
A weather station reports temperature rounded to the nearest 0.1∘0.1^\circC. Is the recorded variable discrete or continuous? What about the true temperature?

Example 3

medium
A survey records (a) number of pets owned and (b) distance commuted in km. Classify each.

Example 4

easy
Discrete or continuous: the air pressure in a tire.

Example 5

easy
Is the number of students in a class discrete or continuous?

Example 6

medium
For a discrete random variable, the probabilities of all outcomes must sum to what? What is the analogous statement for a continuous variable?

Example 7

medium
Why is time considered continuous even though clocks display discrete seconds?

Example 8

challenge
Explain why probability for a continuous variable assigns 00 to any single exact value, unlike a discrete variable.

Example 9

easy
Classify each quantity as discrete or continuous, and explain: (a) number of students in a class, (b) the height of a student, (c) the number of text messages sent, (d) the temperature in a room.

Example 10

medium
True or false: every finite set of numbers is discrete. Justify briefly.

Example 11

challenge
When modeling a population over time, ecologists sometimes use a discrete recurrence Nt+1=rNt(1−Nt/K)N_{t+1} = rN_t(1 - N_t/K) and other times use a continuous ODE dNdt=rN(1−N/K)\frac{dN}{dt} = rN(1 - N/K). Give one reason each model is preferred in its context.

Example 12

medium
A dataset of children's shoe sizes is plotted with a histogram. Why is a bar chart with gaps actually more honest than a histogram here?

Example 13

easy
Classify as discrete or continuous: the length of a fish in centimeters.

Example 14

hard
Give a concrete example where treating a discrete variable as continuous gives a wrong probability statement.

Example 15

hard
Two ways to plot exam scores out of 100: dot plot of integer scores, or histogram with bins of width 10. Which respects discreteness, and what does the histogram trade away?

Example 16

medium
Number of defective items in a batch of 100: discrete or continuous, and what are the possible values?

Example 17

hard
Classify each as discrete or continuous: (a) the integers Z\mathbb{Z}, (b) the interval [0,1][0, 1], (c) the set {1/n:n∈Z+}\{1/n : n \in \mathbb{Z}^+\}.

Example 18

easy
A quantity whose possible values are isolated points is called ___.

Example 19

medium
Classify each: (a) number of phone calls received per day, (b) battery voltage in volts.

Example 20

hard
Argue why money (in dollars and cents) is best modeled as discrete for accounting but often as continuous in financial models.
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