Exercise 12.4: Probability - Tree Diagrams, Possibility Grids & Independent Events

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Chapter Overview & SLOs

What is a possibility diagram and how do you use it to calculate probability? A possibility diagram (also called a sample space grid) organizes all joint combinations of two-stage experiments along intersecting coordinate axes. This visual method helps identify total outcomes and favorable outcomes for probability calculations.

Possibility Diagram Construction Steps:

1. List outcomes of Trial 1 as rows
2. List outcomes of Trial 2 as columns
3. Fill each cell with the combined result (sum, product, etc.)
4. Count total cells = $n \times m$ (where $n$ and $m$ are number of outcomes in each trial)
5. Count favorable cells matching your condition
6. Apply probability formula: $P(E) = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}}$

Example: Card Drawing with Replacement

Cards numbered 1, 2, and 3 are drawn in two consecutive trials (with replacement):

Sample space grid showing product of the two numbers:

$$\begin{array}{c|ccc} \times & 1 & 2 & 3 \\\hline 1 & 1 & 2 & 3 \\ 2 & 2 & 4 & 6 \\ 3 & 3 & 6 & 9 \end{array}$$

Total outcomes: $3 \times 3 = 9$

Favorable outcomes (even product): $\{2, 2, 4, 6, 6\}$ = 5 outcomes

Probability of even product: $P(\text{Even}) = \frac{5}{9}$

What is a probability tree diagram and when do you use it? A tree diagram is used for sequential events with 3 or more stages. Each branch represents a possible outcome with its probability.

Coin Toss Tree Diagram Example:

Three consecutive coin flips, each with $P(H) = 0.5$ and $P(T) = 0.5$:

• Total outcomes: $2^3 = 8$ terminal paths
• Sample space: $\{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$
• Probability of $TTT$: $P(TTT) = \frac{1}{8}$
• Probability of exactly 2 heads: Count favorable paths $\{HHT, HTH, THH\}$ = 3 outcomes → $P = \frac{3}{8}$

What is the product rule for independent events? Two events A and B are independent if the outcome of one does not affect the other. The joint probability of both occurring is the product of their individual probabilities:

$$P(A \cap B) = P(A) \times P(B)$$

Medical Diagnostics Example:

• Test A positive probability: $P(A) = 0.9$
• Test B positive probability: $P(B) = 0.7$
• $P(\text{Positive on Both}) = 0.9 \times 0.7 = 0.63$ (63% chance)

Industrial Quality Control Example:

• Pass Quality Check A: $P(A) = 0.9$
• Pass Quality Check B: $P(B) = 0.8$
• $P(\text{Pass Both}) = 0.9 \times 0.8 = 0.72$ (72% chance)

Key Probability Formulas Summary:

• Classical Probability: $P(E) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$
• Independent Events Product Rule: $P(A \cap B) = P(A) \times P(B)$
• Total outcomes in tree diagram: $n^k$ where $n$ = outcomes per trial, $k$ = number of trials
• Total outcomes in possibility grid: $n \times m$ where $n$ and $m$ = outcomes in each trial

What key concepts are tested in FBISE Exercise 12.4? This comprehensive exercise tests:

• Constructing possibility diagrams (sample space grids) for two-stage experiments
• Drawing probability tree diagrams for sequential events
• Identifying favorable outcomes and total outcomes from visual diagrams
• Applying the classical probability formula: $P(E) = \frac{\text{Favorable}}{\text{Total}}$
• Recognizing independent events (outcome of one doesn''t affect the other)
• Applying the product rule: $P(A \cap B) = P(A) \times P(B)$
• Real-world applications in medical diagnostics and quality control

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do you construct comprehensive sample space possibility grids to map and calculate probabilities for two-stage experiments? Students will create $n \times m$ grids and identify favorable outcomes.
• How do you design sequential probability tree diagrams to track multi-stage experiments like consecutive coin flips? Students will draw branching diagrams and determine probabilities of specific sequences like $TTT$.
• How do you identify independent events and apply the product rule formula $P(A \cap B) = P(A) \times P(B)$ to compute joint intersection probabilities? Students will multiply individual probabilities correctly.
• How do you calculate real-world event probabilities within manufacturing quality control limits and medical screening scenarios? Students will apply probability concepts to practical situations.

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