IB MYP 4-5 Maths- Mutually exclusive events- Study Notes - New Syllabus
IB MYP 4-5 Maths- Mutually exclusive events – Study Notes
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- Mutually exclusive events
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Mutually Exclusive Events
Mutually Exclusive Events
Two or more events are called mutually exclusive (or disjoint) if they cannot occur at the same time. In other words, the occurrence of one event means the other cannot happen in the same trial or experiment.
Example Meaning: If you roll a die, getting a “2” and a “5” on the same roll is impossible.
These outcomes are mutually exclusive.
Key Characteristics of Mutually Exclusive Events:
- No overlap in outcomes: \( A \cap B = \emptyset \)
- Cannot happen simultaneously
- Their probabilities are simply added: \( P(A \cup B) = P(A) + P(B) \)
- \( P(A \cap B) = 0 \)
Comparison: Mutually Exclusive vs. Independent Events
| Aspect | Mutually Exclusive | Independent(Non – Mutually Exclusive) |
|---|---|---|
| Occurrence | Cannot occur together | Can occur together |
| Formula | \( P(A \cup B) = P(A) + P(B) \) | \( P(A \cap B) = P(A) \cdot P(B) \) |
| Intersection | \( P(A \cap B) = 0 \) | \( P(A \cap B) \neq 0 \) |
Real-Life Examples of Mutually Exclusive Events:
- Flipping a coin: Getting heads or tails (not both at once)
- Rolling a die: Getting a 2 or a 5 (you can only get one outcome)
- Selecting a student: A student cannot be both “present” and “absent” on the same day
Example:
Two events are defined as follows: A = rolling an even number on a die, B = rolling an odd number on a die. Are A and B mutually exclusive? Find \( P(A \cup B) \).
▶️Answer/Explanation
Step 1: A = {2, 4, 6}, B = {1, 3, 5}
Step 2: \( A \cap B = \emptyset \), so they are mutually exclusive
Step 3: \( P(A) = \frac{3}{6} = 0.5 \), \( P(B) = \frac{3}{6} = 0.5 \)
Step 4: \( P(A \cup B) = P(A) + P(B) = 1 \)
Answer: Yes, mutually exclusive. \( P(A \cup B) = 1 \)
Example:
A card is drawn from a standard deck. Let A = “drawing a heart”, and B = “drawing a spade”. Are A and B mutually exclusive? What is \( P(A \cup B) \)?
▶️Answer/Explanation
Step 1: A and B are different suits. No card can be both a heart and a spade.
Step 2: Mutually exclusive: \( A \cap B = \emptyset \)
Step 3: \( P(A) = \frac{13}{52} = 0.25 \), \( P(B) = \frac{13}{52} = 0.25 \)
Step 4: \( P(A \cup B) = 0.25 + 0.25 = 0.5 \)
Answer: Yes, they are mutually exclusive. \( P(A \cup B) = 0.5 \)
Example:
In a class of 30 students: 12 like chess, 10 like football, 5 like both. Are the events “liking chess” and “liking football” mutually exclusive?
▶️Answer/Explanation
Step 1: Some students like both sports → overlap exists
Step 2: \( A \cap B \neq \emptyset \)
Answer: No, the events are not mutually exclusive.