IB MYP 4-5 Maths- Combined events- Study Notes - New Syllabus
IB MYP 4-5 Maths- Combined events – Study Notes
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- Combined events
IB MYP 4-5 Maths- Combined events – Study Notes – All topics
Combined Events
Combined Events
Combined events occur when two or more events happen together or in sequence in a single experiment. These events can be considered together to calculate probabilities.
Types of Combined Events:
Union of Events (A or B): Event A happens OR Event B happens (or both).
Formula: \( P(A \cup B) = P(A) + P(B) – P(A \cap B) \)
Intersection of Events (A and B): Both Event A and Event B occur.
Formula: \( P(A \cap B) \)
Complement of an Event: The event does NOT occur.
Formula: \( P(A’) = 1 – P(A) \)
Key Notes:
- If A and B are mutually exclusive, then \( P(A \cap B) = 0 \) and \( P(A \cup B) = P(A) + P(B) \).
- If A and B are independent, then \( P(A \cap B) = P(A) \times P(B) \).
Example:
A card is drawn from a standard deck. Find the probability that it is a heart or a face card.
▶️Answer/Explanation
Step 1: A = heart cards (13 cards), B = face cards (Jack, Queen, King in all suits = 12 cards)
Step 2: Intersection: cards that are both heart and face = 3 (J♥, Q♥, K♥)
Step 3: \( P(A) = \frac{13}{52}, P(B) = \frac{12}{52}, P(A \cap B) = \frac{3}{52} \)
Step 4: \( P(A \cup B) = \frac{13}{52} + \frac{12}{52} – \frac{3}{52} = \frac{22}{52} = \frac{11}{26} \)
Answer: \( \frac{11}{26} \).
Example:
A die is rolled twice. Find the probability of getting a 4 on the first roll and an even number on the second roll.
▶️Answer/Explanation
Step 1: A = getting 4 on first roll = \( \frac{1}{6} \)
B = getting even number on second roll = \( \frac{3}{6} = \frac{1}{2} \)
Step 2: These are independent events, so \( P(A \cap B) = P(A) \times P(B) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \)
Answer: \( \frac{1}{12} \).
Example:
In a group of 50 students, 30 play football (F), 20 play basketball (B), and 10 play both. Find the probability that a student selected plays football or basketball.
▶️Answer/Explanation
Step 1: Use union formula: \( P(F \cup B) = P(F) + P(B) – P(F \cap B) \)
Step 2: \( P(F) = \frac{30}{50} = 0.6,\ P(B) = \frac{20}{50} = 0.4,\ P(F \cap B) = \frac{10}{50} = 0.2 \)
Step 3: \( P(F \cup B) = 0.6 + 0.4 – 0.2 = 0.8 \)
Answer: 0.8 or 80%.