Probability of Compound Events

Definition: A compound event is an event whose sample space is formed by combining the outcomes of two or more individual experiments.

Key facts:

• If experiment 1 has m outcomes and experiment 2 has n outcomes, the combined sample space has m × n outcomes.
• Each combined outcome is an ordered pair (or tuple) of individual outcomes.
• The probability of a compound event = (Number of favourable combined outcomes) / (Total combined outcomes).
• All combined outcomes are assumed equally likely in Class 10 problems.

Tree Diagram Approach:

• A tree diagram represents outcomes as branches from left to right.
• The first level shows outcomes of experiment 1.
• From each outcome of experiment 1, branches show all outcomes of experiment 2.
• Each complete path from root to leaf represents one compound outcome.
• The total number of leaves = total compound outcomes.

Example — Tree for Coin + Die:

• Level 1: H and T (2 branches).
• From each: 1, 2, 3, 4, 5, 6 (6 branches each).
• Total leaves = 2 × 6 = 12 compound outcomes.

Independent Events:

• Two events are independent if the outcome of one does not affect the outcome of the other.
• Rolling two dice: the result on die 1 does not change the probabilities for die 2. They are independent.
• For independent events A and B: P(A and B) = P(A) × P(B).
• This multiplication rule is introduced in Class 10 through compound event problems, though the formal definition comes later.

Mutually Exclusive Events:

• Events that cannot happen simultaneously are mutually exclusive.
• On a single die: "getting 3" and "getting 5" are mutually exclusive (you cannot get both on one roll).
• But on two dice: "die 1 shows 3" and "die 2 shows 5" are NOT mutually exclusive — both can happen together.
• For mutually exclusive events: P(A or B) = P(A) + P(B).

Constructing the Sample Space for Two Dice:

Die 1 can show: {1, 2, 3, 4, 5, 6}. Die 2 can show: {1, 2, 3, 4, 5, 6}.

1. Each outcome is an ordered pair (a, b) where a = die 1 result, b = die 2 result.
2. Total outcomes = 6 × 6 = 36.
3. The sample space can be written as a 6 × 6 grid:

S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),
(2,1), (2,2), (2,3), ..., (2,6),
...
(6,1), (6,2), ..., (6,6)}

Key Subsets (for common problems):

• Sum = 7: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} → 6 outcomes
• Doublets: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} → 6 outcomes
• Sum = 2: {(1,1)} → 1 outcome
• Sum = 12: {(6,6)} → 1 outcome
• Sum ≥ 10: {(4,6), (5,5), (5,6), (6,4), (6,5), (6,6)} → 6 outcomes

Sample Space for Three Coins:

• Coin 1: {H, T}, Coin 2: {H, T}, Coin 3: {H, T}.
• Total = 2 × 2 × 2 = 8 outcomes.
• S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
• Number of heads: 3(×1), 2(×3), 1(×3), 0(×1) — this is the binomial distribution pattern (1, 3, 3, 1).

Sample Space for Coin + Die:

• Coin: {H, T}, Die: {1, 2, 3, 4, 5, 6}.
• Total = 2 × 6 = 12 outcomes.
• S = {(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}

Important observation: The number of outcomes that give a particular sum follows a triangular pattern — increasing from 1 to 6 and then decreasing back to 1. This symmetric distribution is characteristic of the sum of two uniform discrete random variables.

Common Types of Compound Event Problems:

Type 1: Two Dice

• Sample space: 36 outcomes
• Problems ask for P(sum = k), P(doublet), P(product is even), etc.

Type 2: Two or More Coins

• Two coins: 4 outcomes {HH, HT, TH, TT}
• Three coins: 8 outcomes {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Type 3: Coin + Die

• 12 outcomes: {(H,1), (H,2), ..., (H,6), (T,1), ..., (T,6)}

Type 4: Drawing from a Bag (with replacement)

• Two draws with replacement: total outcomes = n², where n = number of items.

Summary of Probabilities for Two Dice:

The distribution is symmetric around 7, which has the highest probability.

Steps to solve compound event problems:

1. Identify the individual experiments and their outcomes.
2. Calculate the total number of combined outcomes using the multiplication principle.
3. List or describe the sample space systematically (grid, tree diagram, or ordered pairs).
4. Define the event E clearly — what condition must the combined outcome satisfy?
5. Count the favourable outcomes (those satisfying the condition).
6. Compute P(E) = favourable / total.

Tools for counting:

• Grid/Table: Best for two dice — list all 36 pairs in a 6×6 table.
• Tree diagram: Best for sequential events (coin then die).
• Systematic listing: Best for small sample spaces (2–3 coins).

Tips for Two-Dice Problems:

• Always write outcomes as ordered pairs (a, b) where a = first die, b = second die.
• (3, 5) and (5, 3) are different outcomes — do not combine them.
• For sum = k, the number of favourable outcomes follows a pattern: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 for sums 2 through 12.
• Sum = 7 has the maximum probability (6/36 = 1/6).

Common Mistakes:

• Confusing ordered and unordered outcomes: When rolling two dice, (3, 5) ≠ (5, 3). Total is 36, not 21.
• Forgetting outcomes: Systematically list using a grid or tree diagram to avoid missing cases.
• Double counting: When finding P(A or B), ensure you don't count outcomes satisfying both A and B twice.

Systematic Grid Method for Two Dice:

Create a 6 × 6 grid with die 1 values (1-6) as rows and die 2 values (1-6) as columns. Each cell represents one outcome. To find P(event), highlight all cells satisfying the condition and count them. This method guarantees no outcomes are missed or double-counted.

Example — Grid for Sums:

• Shade all cells where row + column = 9: cells (3,6), (4,5), (5,4), (6,3) → 4 shaded cells.
• P(sum = 9) = 4/36 = 1/9.

Games and Gambling:

• Board games like Ludo, Monopoly, and Snakes & Ladders use dice. The probability of specific sums determines game strategy.

Quality Control:

• Testing two components together — the combined probability of both passing determines batch acceptance rates.

Weather Forecasting:

• The probability of rain on two consecutive days is a compound event.

Genetics:

• Punnett squares model compound events — combining alleles from two parents to predict offspring traits.

Sports:

• The probability of winning a best-of-3 series requires computing compound probabilities across multiple games.

Insurance:

• Insurers assess the compound probability of multiple independent risk events occurring together to price policies.

Medicine:

• The probability of two independent medical tests both giving false negatives is a compound event calculation crucial for diagnostic reliability.

Cryptography:

• Password strength depends on compound probabilities — the chance of guessing each character correctly in sequence.