IB MYP 4-5 Maths- Probability calculations- Study Notes - New Syllabus
IB MYP 4-5 Maths- Probability calculations – Study Notes
Extended
- Probability calculations
IB MYP 4-5 Maths- Probability calculations – Study Notes – All topics
Probability Calculations
Probability Calculations
Probability measures the likelihood of an event occurring. The value of probability lies between 0 and 1.
Formula:
\( P(E) = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \)
Types of Probability Calculations:
- 1. Simple Probability: Probability of a single event.
- 2. Complementary Probability: Probability that an event does NOT occur.
- 3. Combined Events: Probability of unions and intersections (A or B, A and B).
- 4. Conditional Probability: Probability of one event given another has occurred.
- 5. Independent Events: Events that do not affect each other.
- 6. Mutually Exclusive Events: Events that cannot happen together.
1. Simple Probability
Formula: \( P(E) = \dfrac{\text{Favorable outcomes}}{\text{Total outcomes}} \)
Example : A die is rolled. Find the probability of getting an even number.
▶️Answer/Explanation
Favorable outcomes = {2, 4, 6} → 3 outcomes
Total outcomes = 6
\( P(\text{Even}) = \dfrac{3}{6} = 0.5 \)
Example : A card is drawn from a standard deck of 52 cards. Find the probability it is a heart.
▶️Answer/Explanation
Hearts in a deck = 13
Total cards = 52
\( P(\text{Heart}) = \dfrac{13}{52} = 0.25 \)
2. Complementary Probability
Formula: \( P(\text{Not A}) = 1 – P(A) \)
Example : The probability of rain tomorrow is 0.3. What is the probability it does not rain?
▶️Answer/Explanation
\( P(\text{Not Rain}) = 1 – 0.3 = 0.7 \)
Example : A spinner has equal sections numbered 1–5. Find the probability it does not land on 3.
▶️Answer/Explanation
Total outcomes = 5
P(3) = \( \dfrac{1}{5} \)
P(Not 3) = \( 1 – \dfrac{1}{5} = \dfrac{4}{5} \)
3. Combined Events (Union & Intersection)
Formulas:
- \( P(A \cup B) = P(A) + P(B) – P(A \cap B) \)
- If A and B are mutually exclusive: \( P(A \cup B) = P(A) + P(B) \)
Example : A card is drawn from a deck. Find the probability it is a king or a queen.
▶️Answer/Explanation
P(King) = \( \dfrac{4}{52} \), P(Queen) = \( \dfrac{4}{52} \)
They are mutually exclusive: \( P(K \cup Q) = \dfrac{4}{52} + \dfrac{4}{52} = \dfrac{8}{52} = \dfrac{2}{13} \)
Example : The probability a student studies Math = 0.6, English = 0.5, both = 0.3. Find P(Math or English).
▶️Answer/Explanation
P(M ∪ E) = P(M) + P(E) – P(M ∩ E)
= 0.6 + 0.5 – 0.3 = 0.8
4. Conditional Probability
Formula: \( P(A|B) = \dfrac{P(A \cap B)}{P(B)} \)
Example : A bag has 3 red and 2 blue balls. One is drawn and kept aside. Find P(next ball is red given first was red).
▶️Answer/Explanation
After first red is removed: Red = 2, Total = 4
P(Red | Red) = \( \dfrac{2}{4} = 0.5 \)
Example : The probability of passing Math = 0.7, passing Science = 0.6, passing both = 0.5. Find P(Science | Math).
▶️Answer/Explanation
P(S | M) = P(S ∩ M) / P(M) = 0.5 / 0.7 ≈ 0.714
5. Independent Events
Rule: If A and B are independent, then \( P(A \cap B) = P(A) \cdot P(B) \).
Example : Toss a coin and roll a die. Find probability of getting heads and a 4.
▶️Answer/Explanation
P(H) = 1/2, P(4) = 1/6
P(H and 4) = (1/2)(1/6) = 1/12
Example : Probability of raining on Saturday = 0.3, Sunday = 0.4 (independent). Find probability it rains both days.
▶️Answer/Explanation
P(Both) = 0.3 × 0.4 = 0.12