9.3 Mutually Exclusive Events and Non-Mutually Exclusive Events
In probability, two events may or may not be able to happen at the same time. When two events cannot occur simultaneously, they are called mutually exclusive events, meaning there is no common outcome or intersection between them (\(A \cap B = \phi\)). Conversely, if two events can occur at the same time and share at least one common outcome (\(A \cap B \neq \phi\)), they are known as non-mutually exclusive events. Understanding this difference is essential for applying the correct addition rule of probability when calculating combined events using Venn diagrams or tree diagrams.
Learning Objectives
- Differentiate between mutually exclusive and non-mutually exclusive events.
- Verify the formula of probability of combined events (the addition rule of probability) for mutually exclusive and non-mutually exclusive events.
- Determine the probability of combined events for mutually exclusive and non-mutually exclusive events using Venn diagrams, tree diagrams, or formulae.
Comparison of Mutually Exclusive vs Non-Mutually Exclusive Events
Show how set intersections determine which addition formula of probability to apply.
Mutually Exclusive Events
Non-Mutually Exclusive Events
Brief Explanation
Mutually Exclusive Events
Events that cannot occur together at the same time. There is no intersection between the sets: \(A \cap B = \phi\) (empty set).
Example: Rolling a fair dice where \(P\) is getting a number greater than 4 \(\{5, 6\}\) and \(R\) is getting a perfect square \(\{1, 4\}\). Since \(P \cap R = \phi\), events \(P\) and \(R\) are mutually exclusive.
Non-Mutually Exclusive Events
Events that can occur together at the same time. There is at least one common outcome: \(A \cap B \neq \phi\).
Example: Rolling a fair dice where \(P\) is getting a number greater than 4 \(\{5, 6\}\) and \(Q\) is getting an even number \(\{2, 4, 6\}\). Since \(P \cap Q = \{6\} \neq \phi\), events \(P\) and \(Q\) are non-mutually exclusive.
The Addition Rule of Probability
The probability of combined events involving "or" (\(A \text{ or } B\), denoted by \(A \cup B\)) depends on whether they are mutually exclusive:
For mutually exclusive events: \[P(A \cup B) = P(A) + P(B)\] This holds because \(P(A \cap B) = 0\).
For non-mutually exclusive events: \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] Here, \(P(A \cap B) \neq 0\), so the intersection probability is subtracted once to avoid double-counting.
Notation summary: \(P(A \text{ and } B) = P(A \cap B)\) and \(P(A \text{ or } B) = P(A \cup B)\).
Try Answering First
Read the question, answer it in your mind, then press "Check Answer".
A fair dice is rolled. \(Q\) is the event of getting an even number, and \(R\) is the event of getting a perfect square. Determine whether \(Q\) and \(R\) are mutually exclusive or non-mutually exclusive events.
Five cards labelled with the letters "C, I, N, T, A" are placed in a box. A card is drawn at random. Calculate the probability that the card chosen is labelled with a consonant or the letter "A".
The probabilities that Zalifah and Maran eat cendol at a banquet are \(\dfrac{5}{7}\) and \(\dfrac{3}{5}\) respectively. Assuming their choices are independent, calculate the probability that both Zalifah and Maran eat cendol.
Common Mistakes
- Forgetting to subtract \(P(A \cap B)\) when events are non-mutually exclusive, resulting in an incorrect probability that double-counts the common outcomes.
- Assuming two events are always mutually exclusive simply because they are described separately in the question.
- Confusing the terms: writing \(P(A \cap B)\) when calculating "or" (\(A \cup B\)), or writing \(P(A \cup B)\) when looking for common intersection outcomes ("and").
Misconceptions
Students often believe mutually exclusive events mean independent events.
Mutually exclusive events cannot occur at the same time (\(A \cap B = \phi\), \(P(A \cap B) = 0\)). Independent events are events where the occurrence of one event does not affect the occurrence of the other (\(P(A \cap B) = P(A) \times P(B)\)).
Thinking that any two subsets of a sample space must overlap.
Subsets can be completely disjoint (such as odd and even numbers on a dice), meaning they have no overlap at all.
Summary
- Two events \(A\) and \(B\) are mutually exclusive if \(A \cap B = \phi\) (cannot happen together).
- Two events \(A\) and \(B\) are non-mutually exclusive if \(A \cap B \neq \phi\) (can happen together).
- Addition Rule: For mutually exclusive events: \(P(A \cup B) = P(A) + P(B)\). For non-mutually exclusive events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).
- Probability problems can be represented and verified systematically using list of outcomes, Venn diagrams, or tree diagrams.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 Eight cards labelled 4, 5, 6, 7, 8, 9, 10, 11 are placed in a box. A card is chosen at random. Event \(A\) is getting a number greater than 8 and Event \(C\) is getting an even number. Which statement is correct?
2 Two fair coins are tossed simultaneously. Event \(J\) is getting two tails and Event \(K\) is getting two heads. What is the value of \(P(J \text{ or } K)\)?
B. Fill in the Blanks
3 A combined event \(A\) and \(B\) is known as a mutually exclusive event if there is no intersection between them, such that \(A \cap B =\) .
4 For non-mutually exclusive events, the addition rule of probability is \(P(A \cup B) = P(A) + P(B) -\) .
C. Drag and Drop: Match Each Event Description or Formula
Match each event description or formula to its correct classification or mathematical representation. On a phone, tap the answer choice first, then tap the destination box.
