IB MYP 4-5 Maths- Dependent and independent events- Study Notes - New Syllabus

IB MYP 4-5 Maths- Dependent and independent events – Study Notes

Extended

Dependent and Independent Events

Definition of Event: An event is an outcome or a set of outcomes from a probability experiment.

Independent Events:

Two events A and B are independent if the occurrence of one does not affect the occurrence of the other.

Example: Tossing a coin and rolling a die.

Condition: $ P(A|B) = P(A) $ and $ P(B|A) = P(B) $

Multiplication Rule: $ P(A \cap B) = P(A) \times P(B) $

Dependent Events:

Two events A and B are dependent if the occurrence of one affects the occurrence of the other.

Example: Removing colored marbles from a bag. Each time you remove a marble the chances of drawing out a certain color will change.

Condition: $ P(A|B) \neq P(A) $

Multiplication Rule: $ P(A \cap B) = P(A) \times P(B|A) $

Comparison Table:

Feature	Independent Events	Dependent Events
Effect	One does not affect the other	One affects the other
Formula	$ P(A \cap B) = P(A)P(B) $	$ P(A \cap B) = P(A)P(B\|A) $
Example	Toss coin, roll die	Pick 2 cards without replacement

Formulas Summary:

Example:

A coin is tossed and a die is rolled. Find the probability of getting heads and a 4.

▶️Answer/Explanation

P(Heads) = $ \dfrac{1}{2} $

P(4) = $ \dfrac{1}{6} $

P(Heads ∩ 4) = $ \dfrac{1}{2} \times \dfrac{1}{6} = \dfrac{1}{12} $

Example:

Two cards are drawn without replacement from a standard deck. Find the probability both are aces.

▶️Answer/Explanation

P(First ace) = $ \dfrac{4}{52} $

P(Second ace | first ace) = $ \dfrac{3}{51} $

P(Both aces) = $ \dfrac{4}{52} \times \dfrac{3}{51} = \dfrac{12}{2652} = \dfrac{1}{221} $

Example:

If $P(A) = 0.5, P(B) = 0.4$ and $P(A ∩ B) = 0.2$, check if A and B are independent.

▶️Answer/Explanation

For independence,$ P(A ∩ B)$ should equal $P(A) × P(B).$

$P(A) × P(B) = 0.5 × 0.4 = 0.2$

$P(A ∩ B) = 0.2$ (same)

So, A and B are independent.