Probability of two events

Overview

Probability measures how likely an event is to happen. A probability is always between $0$ and $1$:

$$ 0 \le P(\text{event}) \le 1 $$

A probability of $0$ means the event is impossible. A probability of $1$ means the event is certain. Most real questions sit between these two ends.

This chapter builds the bridge from one-event probability to two-event probability. A learner first learns to describe the sample space, count favourable outcomes, and write a probability as:

$$ P(\text{event})=\frac{\text{number of favourable outcomes}}{\text{number of possible outcomes}} $$

Then the learner extends that idea to questions involving two actions, two conditions, or two events in the same situation. The main habit is simple but powerful: identify exactly what is being selected, count the total outcomes, then count only the outcomes that match the event.

Prerequisites

• Sets, subsets, operations with sets, and Venn diagrams of two sets - Probability uses the language of sets, including union, intersection, complement, "only", "both", and "neither".
• Ratios and proportions - A probability is a ratio comparing favourable outcomes with all possible outcomes.
• Repeating decimals and fractions - Probabilities may be written as fractions, decimals, or percentages.
• Frequency distribution - Frequency tables can be used to form probabilities from observed data.
• Approximations, rounding, significant figures, and decimal places - Some probability answers need rounding to a stated number of decimal places.

Learning Scope

This chapter covers sample spaces, events, complements, probability from lists and tables, probability from simple two-set information, and probability for two-stage experiments such as choosing clothes, tossing coins, rolling dice, or testing seeds.

The page gives the counting foundation needed before using the formal rules for mutually exclusive events, dependent events, and combined events. Those rules are developed further in Mutually exclusive, dependent, and combined events.

This page does not treat probability as advanced statistics. It focuses on the Form IV learner skill of reading a situation carefully, making a fair count, and writing the probability in a clear mathematical form.

Subtopics

Probability Language

An experiment is an action whose result is not known before it happens. Examples include tossing a coin, rolling a die, choosing a student, or selecting a seed.

An outcome is one possible result of the experiment. If a die is rolled, one outcome is $4$.

The sample space is the set of all possible outcomes. For one die:

$$ S=\{1,2,3,4,5,6\} $$

An event is a set of outcomes from the sample space. If $A$ is the event "getting an even number", then:

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

So:

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

Checking routine: Before calculating, ask:

1. What is the experiment?
2. What are all possible outcomes?
3. Which outcomes are favourable?
4. Is each outcome equally likely?

One Event As A Foundation For Two Events

Two-event probability becomes easier when the one-event idea is strong. For example, if a class has $50$ students and $35$ are boys, the event "chosen student is a boy" has:

$$ P(\text{boy})=\frac{35}{50}=\frac{7}{10} $$

This is a one-event probability because only one selection is made and one condition is checked.

If the question changes to "a boy and a girl are selected", "a blue shirt and a black trouser are selected", or "at least two seeds germinate", the learner must track more than one event. The same idea still applies: count favourable outcomes over total outcomes.

Complement Events

The complement of an event is the event that it does not happen. If $A$ is an event, the complement may be written as $A'$.

The two probabilities add to $1$:

$$ P(A)+P(A')=1 $$

So:

$$ P(A')=1-P(A) $$

Example: If the probability that a seed germinates is $\frac{1}{3}$, then the probability that it does not germinate is:

$$ 1-\frac{1}{3}=\frac{2}{3} $$

Warning sign: If a probability and its complement do not add to $1$, at least one count or fraction is wrong.

Probability From A Frequency Table

Sometimes probability is based on observed frequencies. If a table shows $20$ offices and $5$ of them have two tables, then:

$$ P(\text{office has two tables})=\frac{5}{20}=\frac{1}{4} $$

For a phrase such as "at least five tables", include all outcomes that are $5$ or more. If $1$ office has five tables and $2$ offices have six tables, then:

$$ P(\text{at least five tables})=\frac{1+2}{20}=\frac{3}{20} $$

Correction habit: Underline boundary words such as "at least", "at most", "more than", "less than", "only", and "neither". These words decide which entries are counted.

Two Events From A Product Table

When two choices are made from separate groups, list or count the combinations.

Example: A person has $2$ shirts, blue and red, and $3$ trousers, black, green, and yellow. The total number of outfits is:

$$ 2 \times 3 = 6 $$

The sample space can be written as:

$$ \{(B,Blk),(B,G),(B,Y),(R,Blk),(R,G),(R,Y)\} $$

There is only one outfit with a blue shirt and black trouser:

$$ P(\text{blue shirt and black trouser})=\frac{1}{6} $$

Concept bridge: Multiplication here is not a shortcut to memorise blindly. It is counting branches: each shirt can go with each trouser.

Two Events From Repeated Trials

Some questions repeat the same action. For example, three seeds are tested, and each seed may germinate or fail to germinate.

Let $G$ mean germinates and $F$ mean fails. For three seeds, possible patterns include:

$$ GGG,\quad GGF,\quad GFG,\quad FGG,\quad GFF,\quad FGF,\quad FFG,\quad FFF $$

If the probability of germination is $\frac{1}{3}$, then the probability of failure is $\frac{2}{3}$.

The event "at least two seeds germinate" includes:

$$ GGG,\quad GGF,\quad GFG,\quad FGG $$

Each pattern is then calculated using branch probabilities.

Tree Diagrams As A Counting Tool

A tree diagram shows choices or outcomes stage by stage. The branches help the learner avoid missing outcomes.

For two coin tosses, the first toss has $H$ or $T$. From each of those, the second toss has $H$ or $T$:

$$ HH,\quad HT,\quad TH,\quad TT $$

If the question asks for one head and one tail, both $HT$ and $TH$ are favourable. Therefore:

$$ P(\text{one head and one tail})=\frac{2}{4}=\frac{1}{2} $$

Warning sign: Do not count only $HT$ and forget $TH$. In two-stage experiments, order may create different outcomes.

Venn Diagram Probability

Two-set information can become probability when a person or item is selected at random.

If $35$ people are surveyed, $18$ keep goats, $20$ keep cows, and $3$ keep both, then the number who keep goats only is:

$$ 18-3=15 $$

So the probability that a randomly selected person keeps goats only is:

$$ \frac{15}{35}=\frac{3}{7} $$

The probability step comes after the set-counting step. First find the correct region, then divide by the total number in the sample.

Fairness And Equally Likely Outcomes

The formula:

$$ P(A)=\frac{n(A)}{n(S)} $$

works directly when the outcomes are equally likely. A fair die has equally likely faces. A class list may also be equally likely if each student has the same chance of being selected.

But if some outcomes have different chances, use the given probabilities rather than only counting labels. For example, if a seed has probability $\frac{1}{3}$ of germinating, do not treat $G$ and $F$ as equally likely just because there are two labels. Their probabilities are $\frac{1}{3}$ and $\frac{2}{3}$.

Key Terms

| Term | Meaning | | --- | --- | | Experiment | An action or process with an uncertain result. | | Outcome | One possible result of an experiment. | | Sample space | The set of all possible outcomes. | | Event | A set of outcomes being considered. | | Favourable outcome | An outcome that satisfies the event. | | Complement | The event that the chosen event does not happen. | | Random selection | A selection where each item in the stated group has an equal chance unless told otherwise. | | Tree diagram | A stage-by-stage diagram for listing outcomes and branch probabilities. | | At least | The stated number or more. | | At most | The stated number or less. |

Worked Examples

Example 1: Probability From A Class

A class has $50$ students. There are $35$ boys and $15$ girls. If one student is chosen at random, find the probability that the student is a boy.

Step 1: Identify the total.

There are $50$ students.

Step 2: Count the favourable outcomes.

There are $35$ boys.

Step 3: Write the probability.

$$ P(\text{boy})=\frac{35}{50} $$

Step 4: Simplify.

$$ \frac{35}{50}=\frac{7}{10} $$

So the probability is $\frac{7}{10}$.

Check: The answer is less than $1$ and greater than $\frac{1}{2}$, which makes sense because boys are more than half the class.

Example 2: Blue Shirt And Black Trouser

Jonika has $2$ shirts: blue and red. She has $3$ trousers: black, green, and yellow. If one shirt and one trouser are chosen at random, find the probability of choosing a blue shirt and a black trouser.

Step 1: Count all outfits.

$$ 2 \times 3 = 6 $$

Step 2: Count favourable outfits.

Only one outfit is blue shirt and black trouser.

Step 3: Write the probability.

$$ P(\text{blue and black})=\frac{1}{6} $$

Check: A tree diagram would show six final branches, and one branch matches the event.

Example 3: At Least Two Seeds Germinate

A farmer tests three seeds. The probability that a seed germinates is $\frac{1}{3}$. Find the probability that at least two seeds germinate.

Step 1: Find the complement probability for one seed.

Let $G$ mean germinates and $F$ mean fails.

$$ P(G)=\frac{1}{3},\qquad P(F)=\frac{2}{3} $$

Step 2: List the favourable patterns.

"At least two" means two or three seeds germinate:

$$ GGG,\quad GGF,\quad GFG,\quad FGG $$

Step 3: Calculate each pattern.

$$ P(GGG)=\frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}=\frac{1}{27} $$

Each pattern with two germinations and one failure has probability:

$$ \frac{1}{3}\times \frac{1}{3}\times \frac{2}{3}=\frac{2}{27} $$

There are three such patterns:

$$ GGF,\quad GFG,\quad FGG $$

Step 4: Add the favourable probabilities.

$$ P(\text{at least two germinate})=\frac{1}{27}+3\left(\frac{2}{27}\right)=\frac{7}{27} $$

So the probability is $\frac{7}{27}$.

Check: The answer is smaller than $\frac{1}{2}$, which is reasonable because each seed is more likely to fail than germinate.

Example 4: Probability From A Table

The number of tables in $20$ offices is shown below.

| Number of tables | 1 | 2 | 3 | 4 | 5 | 6 | | ---: | ---: | ---: | ---: | ---: | ---: | ---: | | Number of offices | 4 | 5 | 6 | 2 | 1 | 2 |

Find the probability that an office selected at random has at least five tables.

Step 1: Interpret the phrase.

"At least five" means $5$ or $6$ tables.

Step 2: Count favourable offices.

$$ 1+2=3 $$

Step 3: Count total offices.

$$ 4+5+6+2+1+2=20 $$

Step 4: Write the probability.

$$ P(\text{at least five tables})=\frac{3}{20} $$

Common Mistakes

| Mistake | Why It Is A Problem | Correction | | --- | --- | --- | | Treating every named outcome as equally likely | $G$ and $F$ may not have the same probability. | Use the given branch probabilities when they are provided. | | Forgetting the total sample size | The denominator becomes wrong. | Find the full sample space before counting favourable cases. | | Misreading "at least" | Learners may count only the exact value. | "At least $5$" means $5$ or more. | | Misreading "at most" | Learners may count values above the limit. | "At most $5$" means $5$ or less. | | Counting "blue then black" but ignoring order in similar problems | Some two-stage outcomes have more than one arrangement. | List the outcomes or draw a tree before adding probabilities. | | Giving an answer greater than $1$ | A probability cannot exceed certainty. | Recheck numerator and denominator; favourable outcomes cannot exceed total outcomes. | | Finding a Venn region but not converting it to probability | The answer remains a count, not a probability. | Divide the correct region by the total number selected from. |

Practice Tasks

Foundation

1. A die is rolled once. Write the sample space.
2. A die is rolled once. Find the probability of getting an odd number.
3. A bag has $4$ red balls and $6$ blue balls. One ball is selected at random. Find $P(\text{red})$.
4. If $P(A)=\frac{3}{8}$, find $P(A')$.
5. A class has $24$ girls and $16$ boys. Find the probability that a randomly selected learner is a girl.

Skill-Building

1. A person has $3$ shirts and $2$ pairs of trousers. How many different outfits can be formed?
2. A fair coin is tossed twice. List the sample space.
3. A fair coin is tossed twice. Find the probability of getting exactly one head.
4. A table shows that $8$ out of $40$ households use solar lamps. Find the probability that a randomly selected household uses a solar lamp.
5. In a group of $60$ learners, $25$ study Biology and $12$ study both Biology and Chemistry. Find the probability that a learner selected at random studies Biology only.

Exam-Style

1. A student has $2$ shirts, black and white, and $4$ skirts, blue, red, green, and yellow. Draw a tree diagram and find the probability of choosing a white shirt and a red skirt.
2. The probability that a seed germinates is $\frac{2}{5}$. Three seeds are planted. Find the probability that all three seeds germinate.
3. The probability that a seed germinates is $\frac{2}{5}$. Three seeds are planted. Find the probability that exactly two seeds germinate.
4. A frequency table shows that $6$ workers earn below Tsh $200000$, $14$ earn from Tsh $200000$ to Tsh $399999$, and $10$ earn Tsh $400000$ or more. Find the probability that a worker selected at random earns at least Tsh $400000$.
5. In a sample of $35$ people, $18$ keep goats, $20$ keep cows, and $3$ keep both. Find the probability that a person selected at random keeps goats only.

Challenge

1. A fair die is rolled twice. Find the probability that the sum is $7$.
2. A fair die is rolled twice. Find the probability that both numbers are even.
3. A seed has probability $\frac{1}{3}$ of germinating. Four seeds are planted. Find the probability that at least three seeds germinate.
4. A survey has $80$ people. $45$ use mobile money, $30$ use bank cards, and $18$ use both. Find the probability that a randomly selected person uses exactly one of the two services.
5. Create your own two-stage probability question and solve it by listing the sample space.

Generated Question Layer

• Conceptual questions: Ask learners to define experiment, outcome, sample space, event, complement, and favourable outcome.
• Counting questions: Generate sample spaces for coins, dice, clothes, classroom selections, and simple table contexts.
• Complement questions: Give $P(A)$ and ask for $P(A')$, including fractions with unlike denominators.
• Table questions: Generate frequency tables and ask for exact, at least, at most, more than, and less than probabilities.
• Tree diagram questions: Use two-stage and three-stage contexts where branches have equal or unequal probabilities.
• Venn bridge questions: Ask for a set region first, then convert it into a probability.
• Edge cases: Include impossible events, certain events, repeated outcomes, boundary words, and answers that require simplification.

Learner Aid Opportunities

• diagram: Show sample space, event, and complement as labelled regions inside a universal set.
• chart: Compare "exactly", "at least", "at most", "more than", and "less than" with example counts.
• interactive: Let learners build a tree diagram by choosing the number of stages and branch probabilities.
• animation: Reveal outcomes one branch at a time so learners see why multiplication counts combinations.
• video: Model how to read a probability word problem before calculating.
• LLM tutor: Ask guiding questions: "What is the total?", "Which outcomes are favourable?", and "Does the answer lie between 0 and 1?"

Exam-Derived Signals

The table below is based on unreviewed local extraction and mapping data. It should be treated as a planning signal only, not as an official past-question list. The official 2022 format crosswalk is the only official signal named here, and it groups this topic with Sets/Probability rather than separating every subskill.

| Source | Status | Signal | | --- | --- | --- | | data/exam_format_topic_crosswalk_2022.jsonl | Official format crosswalk | The 2022 examination format maps Sets/Probability to this topic, Sets, subsets, operations with sets, and Venn diagrams of two sets, and Mutually exclusive, dependent, and combined events, with $1$ item and $7.14\%$ weighting. | | data/topic_frequency_2021_2025.json | Unreviewed extraction | topic-probability-of-two-events is counted $5$ times across the 2021-2025 extracted mapping data. | | csee_041_2022_p1_q03_b | Unreviewed question-map signal | A tree-diagram seed-germination context asks for the probability that at least two seeds germinate. | | csee_041_2023_p1_q03_b_i | Unreviewed question-map signal | A random-selection class context asks for the probability that the selected student is a boy. | | csee_041_2023_p1_q03_b_ii | Unreviewed question-map signal | A clothing tree-diagram context asks for the probability of a specified shirt-and-trouser combination. | | csee_041_2024_p1_q03_b_i | Unreviewed question-map signal | A frequency-table context asks for the probability of selecting an office with exactly two tables. | | csee_041_2024_p1_q03_b_ii | Unreviewed question-map signal | A frequency-table context asks for the probability of selecting an office with at least five tables. | | csee_041_2021_p1_q03_b_ii; csee_041_2025_p1_q03_b | Unreviewed multi-topic signals | Venn diagram questions include probability-from-region steps and are mapped primarily to sets with this topic as a secondary signal. |

These signals suggest that learners should practise tree diagrams, frequency-table probability, random selection from a group, and probability after a Venn-region count. They must be checked against original papers before being promoted to reviewed past-question 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 40, under the probability and statistics hub.
• Preserved official topic identity: topic-probability-of-two-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, worked examples, practice tasks, corrections, and generated-question prompts are original draft expansion content.
• Review risk: A Mathematics reviewer should check the exact boundary between this topic and Mutually exclusive, dependent, and combined events, especially for tree-diagram and repeated-trial examples.
• Renderer note: This page uses $...$ and $$...$$ math notation for wiki rendering.