IB MYP 4-5 Maths- Probability with Venn diagrams, tree diagrams and sample spaces- Study Notes - New Syllabus
IB MYP 4-5 Maths- Probability with Venn diagrams, tree diagrams and sample spaces – Study Notes
Standard
- Probability with Venn diagrams, tree diagrams and sample spaces
Probability with Venn diagrams, tree diagrams and sample spaces
Sample Space
The sample space of an experiment is the set of all possible outcomes. It is often denoted by \( S \).
Key Points:
- Each outcome is called a sample point.
- Probability of an event = \( \dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \).
- Sample space can be represented by a list, table, or diagram.
Example:
Two dice are rolled. Construct the sample space and find the probability of getting:
- (a) A sum of 7
- (b) A double (same numbers)
▶️Answer/Explanation
Step 1: Sample space size = \( 6 \times 6 = 36 \).
Step 2: Sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes.
\( P(\text{sum of 7}) = \dfrac{6}{36} = \dfrac{1}{6} \).
Step 3: Doubles: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) → 6 outcomes.
\( P(\text{double}) = \dfrac{6}{36} = \dfrac{1}{6} \).
Example:
A bag contains 2 red balls (R), 1 blue (B), and 1 green (G). Two balls are drawn at random with replacement. List the sample space and find the probability that both are red.
▶️Answer/Explanation
Step 1: Sample space = {(R,R), (R,B), (R,G), (B,R), (B,B), (B,G), (G,R), (G,B), (G,G)} → 9 outcomes.
Step 2: P(R on first draw) = 2/4 = 0.5, P(R on second draw) = 0.5.
Step 3: Both red = \( 0.5 \times 0.5 = 0.25 \).
Tree Diagrams in Probability
A tree diagram is a visual representation of all possible outcomes of an event and their associated probabilities. It is especially useful for compound events and calculating probabilities using the multiplication rule.
Key Points:
- Each branch represents an outcome and its probability.
- The sum of probabilities on branches from a single node = 1.
- To find the probability of a combined event, multiply along the branches.
Multiplication Rule: \( P(A \text{ and } B) = P(A) \times P(B \mid A) \)
Example:
A coin is tossed twice. Use a tree diagram to find the probability of getting:
- (a) Two heads
- (b) One head and one tail
▶️Answer/Explanation
Step 1: Outcomes for first toss: H, T. Outcomes for second toss: H, T.
Step 2: Probabilities for each branch = 0.5.
Step 3:
(a) Two heads = \( 0.5 \times 0.5 = 0.25 \).
(b) One head, one tail = HT or TH = \( 0.25 + 0.25 = 0.5 \).
Example:
A bag contains 3 red and 2 blue balls. Two balls are drawn without replacement. Use a tree diagram to find the probability that both are red.
▶️Answer/Explanation
Step 1: First draw: P(R) = 3/5, P(B) = 2/5.
Step 2: Second draw (without replacement):
If first was red: P(R) = 2/4, P(B) = 2/4.
If first was blue: P(R) = 3/4, P(B) = 1/4.
Step 3: Both red = \( \dfrac{3}{5} \times \dfrac{2}{4} = \dfrac{6}{20} = 0.3 \).
Venn Diagrams in Probability
Venn diagrams visually represent sets and their relationships. In probability, they are used to show sample spaces and calculate probabilities of events, including unions, intersections, and complements.
Key Concepts:
- Union: \( P(A \cup B) = P(A) + P(B) – P(A \cap B) \)
- Intersection: \( P(A \cap B) \) = Probability that both events occur.
- Complement: \( P(A’) = 1 – P(A) \)
- Mutually Exclusive: If \( A \cap B = \emptyset \), then \( P(A \cup B) = P(A) + P(B) \).
Example:
In a group of 100 students:
• 60 like Mathematics (M)
• 45 like Science (S)
• 25 like both
(a) Draw a Venn diagram. (b) Find the probability that a student likes:
- (i) Both subjects
- (ii) Only Mathematics
- (iii) At least one subject
▶️Answer/Explanation
Step 1: Use the formula:
Only Math = \( 60 – 25 = 35 \)
Only Science = \( 45 – 25 = 20 \)
Both = 25
None = \( 100 – (35 + 20 + 25) = 20 \)
Step 2: Probabilities:
- (i) \( P(\text{Both}) = \dfrac{25}{100} = 0.25 \)
- (ii) \( P(\text{Only Math}) = \dfrac{35}{100} = 0.35 \)
- (iii) \( P(\text{At least one}) = 1 – \dfrac{20}{100} = 0.8 \)
Example:
In a survey:
• 40% of people read newspaper A
• 35% read newspaper B
• 15% read both newspapers
Find the probability that a person reads:
- (a) At least one newspaper
- (b) Neither newspaper
▶️Answer/Explanation
Step 1: \( P(A) = 0.40, P(B) = 0.35, P(A \cap B) = 0.15 \).
Step 2: At least one = \( P(A \cup B) = P(A) + P(B) – P(A \cap B) \)
= \( 0.40 + 0.35 – 0.15 = 0.60 \).
Step 3: Neither = \( 1 – 0.60 = 0.40 \).