Determining Probability of Mutually Exclusive Events and Introduction to Independent Events

By the end of this lesson, learners should be able to

1. Define mutually exclusive events and identify them in practical experiments.
2. Apply the addition law P(A or B) = P(A) + P(B) to calculate probabilities of compound mutually exclusive outcomes.
3. Solve multi-context probability problems involving dice, baskets, cards, and number ranges.
4. Distinguish between mutually exclusive events and independent events.

Key vocabulary

Mutually exclusive events

Events that cannot occur at the same time in a single trial.

Rolling a 1 and rolling a 4 on one die are mutually exclusive events.

Addition law

The law stating $P(A\text{or}B)=P(A)+P(B)$ for mutually exclusive events.

Applying the addition law gives $P(\text{red or black})=\frac{8}{20}+\frac{7}{20}=\frac{15}{20}$.

Disjoint sets

Sets that have no common elements, so their intersection is empty.

Multiples of 4 and multiples of 9 between 11 and 30 form disjoint sets.

Independent events

Events where the outcome of one event does not affect the outcome of another.

Tossing a coin and rolling a die are independent events with separate outcomes.

Lesson preview

Slide 1 · Understanding Mutually Exclusive EventsOpening · 10 min

When conducting a single probability trial, such as rolling a die or drawing a single card, two outcomes are mutually exclusive if they cannot happen at the same time. For example, a single roll of a die cannot show both a 2 and a 5 simultaneously. Recording the face that shows up and noting the remaining unshown faces helps us observe that the occurrence of one outcome strictly prevents the simultaneous occurrence of the other.

• In a single experiment, mutually exclusive events cannot occur simultaneously during the same trial.
• Rolling a fair six-sided die once produces exactly one visible top face out of six possible outcomes.
• The set of faces that do not show up forms the complementary set of the observed face.
• Comparing outcome sets confirms whether two events share any common outcomes or are mutually exclusive.

Worked example. 1. Identify outcome set for Event A: $A=\{2\}$. 2. Identify outcome set for Event B: $B=\{5\}$. 3. Find the intersection of A and B: $A\cap B=\mathrm{\varnothing }$ (empty set). 4. Since $A\cap B=\mathrm{\varnothing }$, the two events cannot occur on the same single die roll. Answer: Event $A=\{2\}$ and Event $B=\{5\}$. They share no common outcomes, so they are mutually exclusive..

Check question: A single fair six-sided die is tossed once. Event A is 'getting an even number' and Event B is 'getting a prime number'. Which statement correctly determines whether Events A and B are mutually exclusive?

1. They are not mutually exclusive because rolling a 2 satisfies both events.
2. They are mutually exclusive because they are independent events.
3. They are mutually exclusive because an even number cannot be a prime number.
4. They are not mutually exclusive because rolling a 4 satisfies both events.

Show answer

They are not mutually exclusive because rolling a 2 satisfies both events. The outcomes for Event A (even number) are {2, 4, 6}. The outcomes for Event B (prime number) are {2, 3, 5}. Since the outcome 2 is in both sets, the events can occur at the same time if a 2 is rolled. Therefore, they are not mutually exclusive.

Slide 2 · The Addition Law of ProbabilityModel · 10 min

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• All 4 slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 11 lessons in this sub-strand

See the school projectorWhatsApp us

All 12 lessons in Probability 1

1. 1Performing Probability Experiments and Finding Experimental ProbabilityPerform experiments involving probabilities in different situations
2. 2Empirical Probabilities and Experimental InvestigationsPerform experiments involving probabilities in different situations
3. 3Identifying Range of Probability and Complementary EventsIdentify the range of probability in different situations
4. 4Generating Probability Space of EventsGenerate probability space of different events
5. 5Generating Probability Space for Compound EventsGenerate probability space of different events
6. 6Determining Probability of Mutually Exclusive Events and Introduction to Independent EventsDetermine the probability of mutually exclusive events and independent events
7. 7Determining Probability of Independent Events and Multi-Stage SelectionsDetermine the probability of mutually exclusive events and independent events
8. 8Probability Bounds, Complementary Events, and the Addition LawApply the laws of probability in different situations
9. 9Applying Laws of ProbabilityApply the laws of probability in different situations
10. 10Determining Probability of Independent Events Using Tree DiagramsDetermine the probability of independent events using tree diagrams
11. 11Tree Diagrams for Multi-Stage Independent EventsDetermine the probability of independent events using tree diagrams
12. 12Applications of Probability in Real-Life ContextsAppreciate the application of probability in real-life situations