Probability is the measure of how likely an event is to occur. This topic involves understanding key terms, using formulas, and visualizing outcomes with grids and tree diagrams.

1. Core Concepts (Short Notes)

• Random Experiment: An action where the result is uncertain (e.g., tossing a coin, rolling a die).

• Outcome: A single possible result of an experiment.

• Sample Space (S): The set of all possible outcomes.

• Event (A): A specific outcome or a set of outcomes you are interested in (a subset of the sample space).

• Simple Event: An event with only one outcome.

• Composite Event: An event with more than one outcome.

Key Types of Events

• Mutually Exclusive Events: Events that cannot happen at the same time.

  • Example: When rolling a die, the events "getting an even number" and "getting an odd number" are mutually exclusive.

  • Rule: P(A and B) = 0.

• Independent Events: The outcome of one event does not affect the outcome of another.

  • Example: Tossing a coin and rolling a die. The coin's result has no impact on the die's result.

2. Key Formulas

1. Probability of an Event A: P(A) = Number of favourable outcomes / Total number of outcomes in the sample space P(A) = n(A) / n(S)

2. Complementary Events (Event A NOT happening): P(A') = 1 - P(A)

3. The "OR" Rule (Union):

   • For Mutually Exclusive events: P(A or B) = P(A) + P(B)

   • For events that are Not Mutually Exclusive: P(A or B) = P(A) + P(B) - P(A and B)

4. The "AND" Rule (Intersection for Independent Events): P(A and B) = P(A) × P(B)

3. Visualizing Outcomes: Grids and Tree Diagrams

These tools help you map out the entire sample space for multi-stage experiments.

The Grid Method

• Best for: Two-stage experiments where outcomes are equally likely (e.g., rolling two dice, tossing two coins).

• How it works: Draw a grid. The outcomes of the first experiment label the columns, and the outcomes of the second experiment label the rows. The intersections represent all possible combined outcomes.

Example: Rolling two dice

• Total outcomes = 6 × 6 = 36.

• Probability of the sum being 10: The pairs are (4,6), (5,5), (6,4). There are 3 favourable outcomes. So, P(Sum is 10) = 3/36 = 1/12.

The Tree Diagram Method

• Best for: Multi-stage experiments, especially when events are independent.

• How it works:

  1. Draw branches for the outcomes of the first stage and write the probability on each branch.

  2. From the end of each first-stage branch, draw new branches for the outcomes of the second stage, writing their probabilities.

  3. Multiply along the branches to find the probability of a specific sequence of events.

  4. Add the results of different branches to find the probability of a composite event.

Example: Tossing a coin twice

• The probability of getting "Heads then Tails" (H,T) is P(H) × P(T) = ½ × ½ = ¼.

• The probability of getting exactly one Head is P(H,T) or P(T,H) = P(H,T) + P(T,H) = ¼ + ¼ = ½.

4. Tips & Tricks for Exams

• "At Least One": This phrase is a huge clue. It's often easier to calculate the probability of the opposite event (the complement) and subtract from 1.

  • Question: "Find the probability of getting at least one Head when tossing two coins."

  • Opposite Event: Getting no Heads (i.e., getting Tails on both). P(T,T) = ¼.

  • Solution: P(at least one Head) = 1 - P(no Heads) = 1 - ¼ = ¾.

• With vs. Without Replacement: This concept is crucial for tree diagrams.

  • With Replacement: The probabilities for the second stage stay the same.

  • Without Replacement: The total number of items and the number of the chosen item decrease for the second stage, so the probabilities change.

• Check Your Tree Diagram: The sum of probabilities from any single point on a tree diagram must always add up to 1. The sum of the final probabilities for all possible outcomes must also add up to 1.