Mutually exclusive, dependent, and combined events

Overview

This chapter teaches how probabilities change when events are combined. A learner already knows that one event can be measured by comparing favourable outcomes with all possible outcomes. Here the question becomes sharper:

• Can both events happen together?
• Does the first event change the chance of the second event?
• Does the question use "and", "or", "at least", "only", or "neither"?
• Is the situation best shown by a Venn diagram, a table, or a tree diagram?

Mutually exclusive events cannot happen at the same time. Dependent events are events where one event affects the probability of another. Combined events use probability rules to handle "and", "or", complements, and multi-stage situations.

The aim is not to memorise many formulas separately. The aim is to read the relationship between events, choose the correct representation, and check whether the answer makes sense.

Prerequisites

• Probability of two events - This page depends on sample space, events, complements, tree diagrams, and probability from tables.
• Sets, subsets, operations with sets, and Venn diagrams of two sets - Combined probability uses union, intersection, complement, "only", "both", and "neither".
• Ratios and proportions - Probability operations require accurate fraction multiplication, addition, and simplification.
• Repeating decimals and fractions - Learners must compare and simplify fractions, decimals, and percentages.
• Frequency distribution - Some event probabilities come from tables of observed outcomes.

Learning Scope

This chapter covers mutually exclusive events, non-mutually exclusive events, independent events, dependent events, conditional probability in simple contexts, combined events using "and" and "or", and representations using Venn diagrams, tables, and tree diagrams.

The focus is on Form IV problem solving. Learners should be able to decide whether to add or multiply probabilities, adjust probabilities after a first selection, and check answers against the structure of the question.

This page does not teach advanced probability distributions or formal proof of probability laws. It builds classroom and examination readiness for combined-event questions.

Subtopics

Events That Can Or Cannot Overlap

Two events overlap if they can happen at the same time.

Example: When a die is rolled, the event "even number" and the event "number greater than $3$" overlap because $4$ and $6$ are both even and greater than $3$.

Two events are mutually exclusive if they cannot happen together.

Example: When one die is rolled, the event "getting a $2$" and the event "getting a $5$" are mutually exclusive. A single roll cannot be both $2$ and $5$.

Concept bridge: In set language, mutually exclusive events have no intersection:

$$ A\cap B=\emptyset $$

So:

$$ P(A\cap B)=0 $$

The Addition Rule For Mutually Exclusive Events

If $A$ and $B$ are mutually exclusive, then:

$$ P(A\text{ or }B)=P(A)+P(B) $$

Example: A fair die is rolled once. Find the probability of getting a $2$ or a $5$.

The events cannot happen together, so:

$$ P(2\text{ or }5)=\frac{1}{6}+\frac{1}{6}=\frac{2}{6}=\frac{1}{3} $$

Warning sign: The word "or" often suggests addition, but first check whether the events overlap.

The Addition Rule When Events Overlap

If events can happen together, adding $P(A)$ and $P(B)$ counts the overlap twice. Subtract the overlap once:

$$ P(A\cup B)=P(A)+P(B)-P(A\cap B) $$

Example: A card is selected from numbers $1$ to $10$. Let $A$ be "even" and $B$ be "greater than $6$".

$$ A=\{2,4,6,8,10\} $$

$$ B=\{7,8,9,10\} $$

The overlap is:

$$ A\cap B=\{8,10\} $$

So:

$$ P(A\cup B)=\frac{5}{10}+\frac{4}{10}-\frac{2}{10}=\frac{7}{10} $$

Check: Listing the union gives $\{2,4,6,7,8,9,10\}$, which has $7$ outcomes. The formula agrees.

"And" Usually Means Intersection

The word "and" usually points to an intersection or a path through more than one stage.

For one selection from a set:

$$ P(A\text{ and }B)=P(A\cap B) $$

For two-stage events, "and" usually means follow one branch and then another branch. In those cases probabilities are often multiplied, but the second probability may need adjustment.

Independent Events

Two events are independent if the first event does not change the probability of the second event.

Example: Tossing a coin and rolling a die are independent. The coin result does not change the die result.

If $A$ and $B$ are independent:

$$ P(A\cap B)=P(A)\times P(B) $$

Example: A fair coin is tossed and a fair die is rolled. Find the probability of getting a head and a $6$.

$$ P(H\text{ and }6)=\frac{1}{2}\times \frac{1}{6}=\frac{1}{12} $$

Dependent Events

Two events are dependent if the first event changes the probability of the second event.

A common case is selection without replacement. If one item is removed and not returned, the total number and sometimes the favourable number change.

Example: A bag contains $3$ red balls and $2$ blue balls. Two balls are selected without replacement. Find the probability that both are red.

First red:

$$ P(R_1)=\frac{3}{5} $$

After one red is removed, $2$ red balls remain out of $4$ balls:

$$ P(R_2\mid R_1)=\frac{2}{4} $$

So:

$$ P(R_1\text{ and }R_2)=\frac{3}{5}\times \frac{2}{4}=\frac{3}{10} $$

Warning sign: If the question says "without replacement", do not keep the same denominator.

Conditional Probability In Simple Form

Conditional probability means the probability of one event given that another event has already happened. It may be written as:

$$ P(B\mid A) $$

This reads as "the probability of $B$ given $A$".

For simple dependent selection:

$$ P(A\text{ and }B)=P(A)\times P(B\mid A) $$

The vertical bar does not mean division in this notation. It means "given".

Replacement And No Replacement

Replacement changes the structure of a probability question.

If an item is selected and replaced, the sample space returns to its original size. The trials are usually independent.

If an item is selected and not replaced, the sample space changes. The trials are dependent.

Example: A bag has $4$ green balls and $6$ yellow balls.

With replacement:

$$ P(G\text{ then }G)=\frac{4}{10}\times \frac{4}{10}=\frac{4}{25} $$

Without replacement:

$$ P(G\text{ then }G)=\frac{4}{10}\times \frac{3}{9}=\frac{2}{15} $$

The words of the problem decide which calculation is correct.

Combined Events In Venn Diagrams

Venn diagrams are useful when two events happen inside one population.

For two events $A$ and $B$:

$$ P(A\text{ only})=P(A)-P(A\cap B) $$

$$ P(B\text{ only})=P(B)-P(A\cap B) $$

$$ P(\text{neither})=1-P(A\cup B) $$

If the problem gives counts instead of probabilities, find the counts first and then divide by the total.

Combined Events In Tables

A two-way table can show two categories at once, such as gender and subject choice. To find a probability:

1. Identify the total number in the table.
2. Identify the row, column, or cell required.
3. Divide the favourable count by the total.

For conditional probability from a table, the denominator may change. If the question says "given that the learner is a girl", use the total number of girls as the denominator, not the whole table.

Choosing A Method

Use this decision routine:

| Question Clue | Likely Method | | --- | --- | | One item selected from one group | Count favourable outcomes over total outcomes. | | "Or" with events that cannot overlap | Add probabilities. | | "Or" with events that can overlap | Add probabilities and subtract the overlap. | | "And" in independent stages | Multiply probabilities. | | "And" without replacement | Multiply, but update the second probability. | | "At least" | Use direct listing or use the complement if easier. | | "Given that" | Use conditional probability; change the denominator. | | Two categories in one group | Use a Venn diagram or two-way table. |

Key Terms

| Term | Meaning | | --- | --- | | Mutually exclusive events | Events that cannot happen at the same time. | | Intersection | Outcomes common to two events; written $A\cap B$. | | Union | Outcomes in $A$, in $B$, or in both; written $A\cup B$. | | Independent events | Events where one event does not affect the probability of the other. | | Dependent events | Events where one event changes the probability of the other. | | Conditional probability | Probability of an event after another event is known to have happened. | | Replacement | Returning a selected item before the next selection. | | Without replacement | Not returning a selected item before the next selection. | | Combined event | An event formed by joining simpler events using words such as "and", "or", "at least", or "neither". |

Worked Examples

Example 1: Mutually Exclusive Events

A fair die is rolled once. Find the probability of getting a $1$ or a $6$.

Step 1: Decide whether the events overlap.

One roll cannot be both $1$ and $6$, so the events are mutually exclusive.

Step 2: Add the probabilities.

$$ P(1\text{ or }6)=P(1)+P(6) $$

$$ =\frac{1}{6}+\frac{1}{6}=\frac{2}{6}=\frac{1}{3} $$

So the probability is $\frac{1}{3}$.

Example 2: Events That Overlap

A number is chosen at random from $1$ to $12$. Find the probability that it is even or a multiple of $3$.

Step 1: List the two events.

Even numbers:

$$ \{2,4,6,8,10,12\} $$

Multiples of $3$:

$$ \{3,6,9,12\} $$

Step 2: Find the overlap.

$$ \{6,12\} $$

Step 3: Use the addition rule with overlap.

$$ P(E\cup M)=\frac{6}{12}+\frac{4}{12}-\frac{2}{12}=\frac{8}{12}=\frac{2}{3} $$

Check: The favourable numbers are $\{2,3,4,6,8,9,10,12\}$, which are $8$ out of $12$.

Example 3: Independent Events

A coin is tossed and a die is rolled. Find the probability of getting a tail and an odd number.

Step 1: Identify independence.

The coin result does not affect the die result.

Step 2: Multiply the probabilities.

$$ P(T)=\frac{1}{2} $$

$$ P(\text{odd})=\frac{3}{6}=\frac{1}{2} $$

$$ P(T\text{ and odd})=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4} $$

Example 4: Dependent Events Without Replacement

A box contains $5$ red pens and $3$ blue pens. Two pens are selected without replacement. Find the probability that both pens are blue.

Step 1: First selection.

$$ P(B_1)=\frac{3}{8} $$

Step 2: Update the box.

After selecting one blue pen, $2$ blue pens remain out of $7$ pens.

$$ P(B_2\mid B_1)=\frac{2}{7} $$

Step 3: Multiply along the branch.

$$ P(B_1\text{ and }B_2)=\frac{3}{8}\times \frac{2}{7}=\frac{6}{56}=\frac{3}{28} $$

Example 5: At Least One

A fair coin is tossed three times. Find the probability of getting at least one head.

Step 1: Use the complement.

"At least one head" is the opposite of "no heads".

No heads means:

$$ TTT $$

Step 2: Find the complement probability.

$$ P(TTT)=\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{8} $$

Step 3: Subtract from $1$.

$$ P(\text{at least one head})=1-\frac{1}{8}=\frac{7}{8} $$

Check: Direct listing would include all eight outcomes except $TTT$.

Example 6: Combined Events From A Venn Context

In a group of $80$ learners, $45$ study Physics, $30$ study Chemistry, and $18$ study both. Find the probability that a randomly selected learner studies Physics or Chemistry.

Step 1: Use the union count.

$$ n(P\cup C)=n(P)+n(C)-n(P\cap C) $$

$$ n(P\cup C)=45+30-18=57 $$

Step 2: Convert to probability.

$$ P(P\cup C)=\frac{57}{80} $$

So the probability is $\frac{57}{80}$.

Warning sign: If you add $45+30$ and stop, the $18$ learners who study both have been counted twice.

Common Mistakes

| Mistake | Why It Is A Problem | Correction | | --- | --- | --- | | Adding probabilities for "and" | "And" often means both conditions must happen. | Use intersection or multiply along a branch. | | Multiplying probabilities for "or" | "Or" often means either event is acceptable. | Use addition, then subtract overlap if needed. | | Treating overlapping events as mutually exclusive | The overlap gets counted wrongly. | Check whether both events can happen together. | | Forgetting to subtract the overlap | The same outcomes are counted twice. | Use $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. | | Keeping the same denominator without replacement | The sample space changes after the first selection. | Reduce the total and favourable counts after removal. | | Changing probabilities when replacement is used | Replacement restores the original sample space. | Use the same branch probability after replacement. | | Using the whole sample for a "given that" question | Conditional probability changes the reference group. | Use the group named after "given that" as the denominator. | | Ignoring complement shortcuts | Direct listing may become long. | For "at least one", consider $1-P(\text{none})$. |

Practice Tasks

Foundation

1. Explain why getting a $3$ and getting a $4$ on one die roll are mutually exclusive.
2. A die is rolled once. Find the probability of getting a $1$ or a $2$.
3. A number is chosen from $1$ to $10$. List the outcomes that are even and greater than $5$.
4. If $P(A)=\frac{2}{5}$, $P(B)=\frac{1}{5}$, and $A$ and $B$ are mutually exclusive, find $P(A\cup B)$.
5. State whether tossing a coin and rolling a die are independent or dependent.

Skill-Building

1. A number is chosen from $1$ to $15$. Find the probability that it is a multiple of $2$ or a multiple of $5$.
2. A bag has $6$ red balls and $4$ white balls. Two balls are selected with replacement. Find the probability that both are red.
3. A bag has $6$ red balls and $4$ white balls. Two balls are selected without replacement. Find the probability that both are red.
4. A fair die is rolled twice. Find the probability of getting two even numbers.
5. In a group of $40$ learners, $22$ play football, $18$ play netball, and $7$ play both. Find the probability that a learner plays football or netball.

Exam-Style

1. A box contains $4$ blue pens, $5$ black pens, and $3$ red pens. One pen is selected. Find the probability that it is blue or red.
2. A number is chosen from $1$ to $20$. Find the probability that it is divisible by $3$ or divisible by $4$.
3. A bag contains $5$ green balls and $7$ yellow balls. Two balls are selected without replacement. Find the probability that the first is green and the second is yellow.
4. A coin is tossed three times. Find the probability of getting exactly two heads.
5. In a survey of $100$ people, $55$ use service $A$, $48$ use service $B$, and $20$ use both. Find the probability that a person selected at random uses neither service.

Challenge

1. Two dice are rolled. Find the probability that the sum is $8$ or both dice show the same number.
2. A box contains $3$ red, $4$ blue, and $5$ green beads. Two beads are selected without replacement. Find the probability that they are of different colours.
3. A test has two multiple-choice questions. The probability that a learner answers the first correctly is $\frac{3}{4}$. If the first is correct, the probability that the second is correct is $\frac{2}{3}$. Find the probability that both are correct.
4. In a school, $60\%$ of learners study Agriculture, $45\%$ study Commerce, and $25\%$ study both. Find the probability that a learner studies Agriculture only.
5. Create a two-way table with two categories and write one ordinary probability question and one conditional probability question from it.

Generated Question Layer

• Classification questions: Ask learners to decide whether event pairs are mutually exclusive, overlapping, independent, or dependent.
• Formula-choice questions: Present short contexts and ask which rule applies before any calculation.
• Venn questions: Generate union, intersection, "only", "neither", and complement tasks from two-set counts.
• Tree questions: Generate with-replacement and without-replacement selections, including changing denominators.
• Conditional questions: Use phrases such as "given that", "among those who", and "if the first selected item is".
• Complement questions: Generate "at least one", "none", "at least two", and "not both" tasks.
• Error-diagnosis questions: Show a wrong solution that added, multiplied, or forgot overlap, then ask learners to correct it.

Learner Aid Opportunities

• diagram: Show mutually exclusive circles separated and overlapping events with an intersection region.
• chart: Compare "and", "or", "only", "both", "neither", "given", "with replacement", and "without replacement".
• interactive: Let learners switch between replacement and no replacement and watch branch probabilities change.
• animation: Build the addition rule by adding two regions and then removing the double-counted overlap.
• video: Walk through a Venn probability problem and a tree probability problem side by side.
• LLM tutor: Ask learners to justify the method before calculating: "Can both events happen?", "Did the sample space change?", and "What is the denominator now?"

Exam-Derived Signals

The table below is intentionally conservative. It separates the official format crosswalk from unreviewed extraction signals. No direct reviewed past-question link is claimed for this topic yet.

| Source | Status | Signal | | --- | --- | --- | | data/exam_format_topic_crosswalk_2022.jsonl | Official format crosswalk | The 2022 examination format maps Sets/Probability to this topic, Probability of two events, and Sets, subsets, operations with sets, and Venn diagrams of two sets, with $1$ item and $7.14\%$ weighting. | | data/topic_frequency_2021_2025.json | Unreviewed extraction | topic-mutually-exclusive-dependent-and-combined-events appears in the low-or-no-coverage list, meaning the automatic mapper did not find a clean primary count for recent extracted papers. | | data/question_map_2021_2025.jsonl | Unreviewed extraction | Recent probability signals are mostly mapped to Probability of two events or set/Venn pages, so this topic needs manual review to decide which combined-event questions should also map here. | | Local curriculum sequence | Official registry | This topic follows topic-probability-of-two-events, suggesting it should deepen probability rules after basic two-event counting. |

The absence of clean extracted primary records should not be read as absence from instruction. It means the current automatic mapping did not confidently isolate this more specific topic. A reviewer should audit tree-diagram, Venn-union, replacement, and conditional-probability questions before adding reviewed exam links.

Source And Review Notes

• Official syllabus authority: raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf.
• Registry source: data/curriculum_map.json lists this topic as Form IV, sequence 41, under the probability and statistics hub.
• Preserved official topic identity: topic-mutually-exclusive-dependent-and-combined-events.
• Assessment signal files consulted for planning language: data/topic_frequency_2021_2025.json, data/question_map_2021_2025.jsonl, and data/exam_format_topic_crosswalk_2022.jsonl.
• Learner prose, examples, practice tasks, warnings, and generated-question prompts are original draft expansion content.
• Review risk: The terms "independent", "dependent", and "conditional probability" should be checked against the expected Form IV treatment so the page stays aligned with classroom depth.
• Renderer note: This page uses $...$ and $$...$$ math notation for wiki rendering.