mirror of
https://github.com/HananoshikaYomaru/openfreebooks.git
synced 2026-11-12 20:45:52 +00:00
82 lines
2.8 KiB
Markdown
82 lines
2.8 KiB
Markdown
+++
|
|
title = "Basic probability"
|
|
description = "Calculate theoretical and experimental probabilities for simple events and complements."
|
|
template = "chapter.html"
|
|
weight = 32
|
|
[extra]
|
|
subject = "math"
|
|
chapter_id = "basic-probability"
|
|
strand = "Data Handling"
|
|
+++
|
|
|
|
<div class="book-prose">
|
|
<p>
|
|
Probability measures how likely an event is, from 0 (impossible) to 1 (certain).
|
|
</p>
|
|
|
|
<h2 id="fundamentals" class="book-prose__heading">Fundamentals</h2>
|
|
<p class="book-formula">\[
|
|
P(E)=\frac{\text{number of favorable outcomes}}{\text{number of equally likely outcomes}}.
|
|
\]</p>
|
|
<p>
|
|
Also: \(0\le P(E)\le1\).
|
|
</p>
|
|
|
|
<h2 id="complement" class="book-prose__heading">Complement rule</h2>
|
|
<p class="book-formula">\[
|
|
P(E')=1-P(E).
|
|
\]</p>
|
|
|
|
<h2 id="multiple-events" class="book-prose__heading">Combined events</h2>
|
|
<ul>
|
|
<li>\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)</li>
|
|
<li>If independent: \(P(A\cap B)=P(A)P(B)\)</li>
|
|
</ul>
|
|
|
|
<h2 id="experimental" class="book-prose__heading">Experimental probability</h2>
|
|
<p class="book-formula">\[
|
|
P(E)\approx\frac{\text{observed frequency}}{\text{number of trials}}.
|
|
\]</p>
|
|
|
|
<h2 id="sample-space" class="book-prose__heading">Sample spaces and trees</h2>
|
|
<p>
|
|
Use sample-space tables or tree diagrams for multi-step outcomes.
|
|
Multiply along branches, add across valid branches.
|
|
</p>
|
|
|
|
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
|
<ul>
|
|
<li>Define event clearly before calculating.</li>
|
|
<li>Use fractions first; round at end if needed.</li>
|
|
<li>Check complement and total probability sanity.</li>
|
|
<li>Use diagrams to avoid missing outcomes.</li>
|
|
</ul>
|
|
</div>
|
|
|
|
<div class="book-prose">
|
|
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
|
<details class="book-question">
|
|
<summary class="book-question__prompt">A fair die is rolled. Find \(P(\text{prime})\).</summary>
|
|
<div class="book-question__solution">
|
|
<p><strong>Answer:</strong> \(\frac{3}{6}=\frac12\) (2,3,5).</p>
|
|
</div>
|
|
</details>
|
|
<details class="book-question">
|
|
<summary class="book-question__prompt">A bag has 4 red and 6 blue balls. Find \(P(\text{not red})\).</summary>
|
|
<div class="book-question__solution">
|
|
<p><strong>Answer:</strong> \(\frac{6}{10}=\frac35\).</p>
|
|
</div>
|
|
</details>
|
|
<details class="book-question">
|
|
<summary class="book-question__prompt">For independent events, \(P(A)=0.3,P(B)=0.4\). Find \(P(A\cap B)\).</summary>
|
|
<div class="book-question__solution">
|
|
<p><strong>Answer:</strong> \(0.12\).</p>
|
|
</div>
|
|
</details>
|
|
<details class="book-question">
|
|
<summary class="book-question__prompt">An event occurs 18 times in 50 trials. Estimate probability.</summary>
|
|
<div class="book-question__solution">
|
|
<p><strong>Answer:</strong> \(\frac{18}{50}=0.36\).</p>
|
|
</div>
|
|
</details>
|
|
</div>
|