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Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Probability 7 b i Solution by
Codex 0 2026-10-25
In the uniform balls-in-bins allocation, each ball enters box 1 independently with probability . Thus has the binomial distribution , with probability mass functionand zero outside that range.
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Probability 7 a ii Solution by
Codex 0 2026-10-25
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Probability 7 a i Solution by
Codex 0 2026-10-25
Assume finite second moments and set . By linearity of expectation, centring gives . Expanding its square and using the definitions of variance and covariance,No independence assumption is required for this variance of a sum identity.
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 3 c iii Solution by
Codex 0 2026-10-25
Every nonzero integer has , proving reflexivity. Because , the relation is symmetric. If and , then , proving transitivity. Thus this is an equivalence relation, whose two equivalence classes are the positive integers and the negative integers:
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 3 c ii Solution by
Codex 0 2026-10-25
This binary relation is not an equivalence relation, because it is not symmetric: since , but is false. In fact the injective encoding gives a total order. There are no equivalence classes for this binary relation.
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 3 c i Solution by
Codex 0 2026-10-25
Write . Equality of -values defines an equivalence relation induced by a function: equality is reflexive, symmetric, and transitive, so all three properties hold for . The equivalence class of isFor this is the circle centred at the origin with radius ; the remaining equivalence class is the singleton .
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 3 a v Solution by
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A binary relation is an equivalence relation precisely when it is reflexive, symmetric, and transitive.
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 3 a iv Solution by
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Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 3 a iii Solution by
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Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 3 a ii Solution by
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Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 3 a i Solution by
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Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 2 a iii Solution by
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By the generalized pigeonhole principle, 13 cards guarantee cards in one suit. The guarantee is sharp: 12 cards can consist of three cards from each suit, with no four cards of a common suit.
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 2 a ii Solution by
Codex 0 2026-10-25
Set , using the ceiling function. If no box contains at least objects, all boxes contain at most , so the total number of objects is at most . Taking the contrapositive, a total greater than forces some box to contain at least objects. But , hence . This proves the generalized pigeonhole principle: some box contains at least objects.
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 2 a i Solution by
Codex 0 2026-10-25
The pigeonhole principle states that if objects are placed in boxes and , at least one box contains at least two objects.
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 1 c ii Solution by
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Let contain the multiples of among . Its cardinality is , using the floor function. Because are pairwise coprime, their pairwise intersections correspond to divisibility by , and their triple intersection to divisibility by . The inclusion-exclusion principle gives
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 1 c i Solution by
Codex 0 2026-10-25
For finite sets, the inclusion-exclusion principle statesThus single-set cardinalities are added, pairwise-intersection cardinalities subtracted, triple-intersection cardinalities added, and so on.
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 1 b ii Solution by
Codex 0 2026-10-25
Each of the elements is either included or excluded, so the rule of product gives . Alternatively, partition the power set into disjoint families according to cardinality. The family of -element subsets has size equal to the binomial coefficient . The rule of sum therefore gives
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 1 b i Solution by
Codex 0 2026-10-25
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 1 a iv Solution by
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The rule of product counts unrestricted passwords and passwords containing no A. By the rule of subtraction, the answer is
Past exam of the year 1 of the computer science course of the University of Oxford 2022 Discrete mathematics, continuous mathematics, and probability Discrete mathematics 1 a iii Solution by
Codex 0 2026-10-25
The rule of product says that a sequence of choices, with possibilities at stage regardless of earlier choices, has possible outcomes. Applied to the Cartesian product of finite sets, it gives . Here independence means that the available choices do not restrict one another; no probability assumption is needed.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.Figure 3. Visual Studio Code extension installation.Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. - Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact





