Chapter 15 — Probability

What is Probability? Probability measures how likely an event is to happen. It is a number between 0 and 1 (inclusive), where 0 means the event cannot happen and 1 means it is certain.

Basic terms

• Experiment: Any process with observable outcomes (e.g., toss a coin, roll a die).
• Sample space (S): The set of all possible outcomes of an experiment (e.g., for a die S = {1,2,3,4,5,6}).
• Event (E): Any subset of the sample space (e.g., "getting an even number" = {2,4,6}).
• Equally likely outcomes: Outcomes having the same chance (common assumption for fair coin, fair die, shuffled deck approximations).

Theoretical (classical) probability

If all outcomes are equally likely, the probability of an event E is given by:
P(E) = number of favourable outcomes (n(E)) / total number of outcomes (n(S)).

Experimental (empirical) probability

Based on observed frequency in repeated trials. If an event E occurs f times in N trials, experimental probability ≈ f / N. As N grows large, experimental probability tends to theoretical probability (law of large numbers idea).

Important properties

• 0 ≤ P(E) ≤ 1 for any event E.
• P(S) = 1 and P(∅) = 0 (S = certain event, ∅ = impossible event).
• Complement rule: For event E, its complement E' (event E does not occur) satisfies P(E') = 1 − P(E).
• If A and B are mutually exclusive (cannot happen together), P(A ∪ B) = P(A) + P(B).

How to find probability — simple steps

1. Describe the experiment and list the sample space S.
2. Identify the favourable outcomes for event E and count n(E).
3. Count total outcomes n(S) (assuming equally likely outcomes).
4. Compute P(E) = n(E)/n(S) and simplify if needed.

Short solved examples (classical)

1. Coin toss: S = {H, T}. Probability of getting a head = 1/2.
2. Die roll: S = {1,2,3,4,5,6}. Probability of getting an even number = n(E)=3 → P = 3/6 = 1/2.
3. Bag of 5 balls (2 red, 3 blue): If one ball is drawn at random, P(red) = 2/5.

Relation between theoretical and experimental

Use experimental probability when theoretical calculation is hard or to check theory: repeat the experiment many times, record relative frequency, and compare.

Why this matters (real-life uses)

Probability is used in weather forecasting (chance of rain), games of chance, risk assessment (insurance, finance), quality control, and many decisions under uncertainty.

Random experiment is a process or activity which can be repeated under the same conditions and has one or more possible results, but the exact result cannot be predicted with certainty in advance. Examples: tossing a coin, rolling a die, drawing a card from a shuffled pack.

Outcome (or elementary event) is a single possible result of a random experiment. Each outcome is usually denoted by a symbol such as ω (omega). The sample space S is the set of all possible outcomes of the experiment. For a coin toss S = {Heads, Tails}; for a single die roll S = {1,2,3,4,5,6}.

Key properties: (1) Every outcome of the experiment is an element of S. (2) Outcomes are mutually exclusive: in one trial only one outcome occurs. (3) The sample space is exhaustive: it contains every outcome that can occur.

An event is any subset of the sample space (one or more outcomes). For example, in a die roll, the event “even number” is E = {2,4,6}. When all outcomes are equally likely, probability of an event can be computed by counting outcomes.

Notation summary: S = sample space, ω ∈ S denotes an outcome, E ⊆ S denotes an event, n(S) = number of elements (outcomes) in S, n(E) = number of favorable outcomes for event E.

Definition: A sample space (denoted by S or Ω) of a random experiment is the set of all possible outcomes of that experiment. Each element of S is called an outcome or a sample point.

Notation and examples: If you toss a coin once, S = {Heads, Tails}. If you roll a fair six-sided die once, S = {1, 2, 3, 4, 5, 6}.

Types of sample spaces:

• Finite sample space: has a finite number of outcomes (e.g. a die: 6 outcomes).
• Countably infinite sample space: outcomes can be listed in sequence (e.g. toss a coin until first Head: outcomes = {H, TH, TTH, TTTH, ...}).
• Uncountably infinite (continuous) sample space: outcomes form a continuum (e.g. the exact time in seconds when a bus arrives on an interval [0,60]).

How to represent S: S can be listed as a set (curly braces), described by a rule (set-builder notation), or shown with a tree diagram for sequential experiments. For combined experiments, use the Cartesian product: if experiment A has outcomes S1 and experiment B has outcomes S2, then S = S1 × S2 (list of ordered pairs).

Why S matters: Probability of any event is defined relative to the sample space. For equally likely outcomes, P(event E) = number of favourable outcomes / total number of outcomes = n(E) / n(S).

What is an event? An event is any collection (set) of one or more outcomes of a random experiment. If S is the sample space (set of all possible outcomes), then an event E is a subset of S: E ⊆ S. A single outcome is called a simple (or elementary) event; a set of outcomes is called a compound event.

Common types of events

• Certain event: E = S. It always occurs and P(E) = 1.
• Impossible event: E = Ø. It never occurs and P(E) = 0.
• Complementary event: For event E, the complement E' (or E^c) contains all outcomes in S that are not in E. E and E' are mutually exclusive and exhaustive, and P(E) + P(E') = 1.
• Mutually exclusive (disjoint) events: Two events A and B are mutually exclusive if they cannot occur together, i.e., A ∩ B = Ø. For such events P(A ∪ B) = P(A) + P(B).
• Exhaustive events: A collection of events is exhaustive if their union equals the sample space S (one of them must occur).

Probability of an event (for equally likely outcomes): If n(S) denotes the number of outcomes in the sample space and n(E) the number of favourable outcomes for event E, then

P(E) = n(E) / n(S)

This probability always satisfies 0 ≤ P(E) ≤ 1.

Basic probability rules:

• P(E') = 1 − P(E).
• For any two events A and B: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

Events are represented using set notation and often pictured with Venn diagrams to show intersections, unions and complements. Understanding events helps translate everyday uncertain situations (like tossing a coin, rolling a die, drawing a card) into calculable probabilities.

Definition: When all outcomes of an experiment have the same chance of occurring, they are called equally likely outcomes. The probability of an event is then computed by counting outcomes.

Sample space and outcomes: Let S be the sample space (set of all possible outcomes). If every outcome in S is equally likely, and E is an event (a subset of S), then the probability of E is

P(E) = n(E) / n(S)

where n(E) is the number of outcomes favorable to E and n(S) is the total number of outcomes.

When this applies (classical probability):

• Experiments with symmetry: fair coin, fair die, well-shuffled deck of cards, spinner divided into equal sectors.
• Discrete finite sample spaces where you can list or count outcomes and each outcome is equally likely.

Useful consequences and simple rules:

• Complement rule: P(E') = 1 − P(E).
• If A and B are mutually exclusive events, P(A ∪ B) = P(A) + P(B). (Compute each using counts.)
• For independent experiments with equally likely outcomes (e.g., two fair coin tosses), the sample space size is the product of sizes of individual sample spaces.

Limitations: Not all experiments have equally likely outcomes. If outcomes have different probabilities (biased coin, nonuniform spinner, different-sized balls), you cannot use the simple count formula.

Connection to frequency interpretation: If you repeat an experiment many times, the relative frequency of an event (number of times E occurs divided by total trials) tends to approach the theoretical probability when outcomes are equally likely (law of large numbers).

Definition. If an experiment has a finite number of equally likely outcomes, the probability of an event E is defined as:

P(E) = number of favourable outcomes for E / total number of possible outcomes

Symbolically, if S is the sample space and n(A) denotes the number of elements in a set A, then

P(E) = n(E) / n(S)

Conditions for use:

• The sample space S must be finite.
• All outcomes in S must be equally likely.

How to apply (steps):

1. Describe the sample space S and count n(S).
2. Describe the event E and count n(E) (favourable outcomes).
3. Compute P(E) = n(E)/n(S) and, if needed, simplify the fraction.

Basic properties (derived from the definition):

• 0 ≤ P(E) ≤ 1 for any event E.
• P(S) = 1 and P(∅) = 0.
• If E' is the complement of E, then P(E') = 1 − P(E).
• If A and B are mutually exclusive (disjoint), P(A ∪ B) = P(A) + P(B).
• In general, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

When classical definition is not appropriate: If outcomes are not equally likely or the sample space is infinite, the classical formula n(E)/n(S) does not apply directly; a different (empirical or axiomatic) approach is used.

Probability measures how likely an event is to occur. For any random experiment, we define a sample space S (set of all possible outcomes) and an event A (a subset of S). The probability P(A) is a number between 0 and 1 that quantifies the chance of A happening.

Key basic properties:

• Range: 0 ≤ P(A) ≤ 1 for any event A. If P(A)=0 the event is impossible; if P(A)=1 the event is certain.
• Probability of the sample space: P(S) = 1, because something in S must occur.
• Probability of the empty set: P(∅) = 0.
• Complement rule: For the complement A' (events in S but not in A), P(A) + P(A') = 1. So P(A') = 1 − P(A).
• Addition rule (general): For any two events A and B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). This avoids double counting outcomes in both A and B.
• Mutually exclusive (disjoint) events: If A and B cannot happen together (A ∩ B = ∅), then P(A ∪ B) = P(A) + P(B).
• Monotonicity: If A ⊆ B then P(A) ≤ P(B).

These properties follow from the axioms of probability and are used to compute and reason about probabilities in simple experiments (coin tosses, dice rolls, card draws) and in real-life situations (reliability, weather forecasts, risk assessment).

Definition: Two events A and A' (read “A complement” or “not A”) are complementary if exactly one of them occurs for every outcome in the sample space S. That is, A and A' are mutually exclusive and together exhaustive: A ∩ A' = Ø and A ∪ A' = S.

Notation: A', A^c, or Ā denotes the complement of A (the event “A does not occur”).

Key property (probability): Since A and A' are exhaustive and mutually exclusive, their probabilities add to 1:

• P(A) + P(A') = 1
• so P(A') = 1 − P(A)

Why this holds (intuitive): The probability of something happening (A) plus the probability of it not happening (A') must account for every possibility in S, so the sum is the total probability 1.

When to use: Often it is easier to find P(A') and then get P(A) as 1 − P(A'), especially when A' has fewer or simpler outcomes.

Simple example (die): Let A = “roll an even number” on a fair six-sided die. Outcomes for A: {2,4,6} so P(A) = 3/6 = 1/2. Then P(A') = 1 − 1/2 = 1/2 (A' = {1,3,5}).

What is Experimental (Empirical) Probability?

Experimental probability (also called empirical or relative-frequency probability) is the probability of an event estimated from the results of an experiment or observation. It is found by performing the experiment a number of times, counting how often the event occurs, and dividing by the total number of trials.

Definition and formula

If an event E occurs n(E) times in N repeated trials, the experimental probability of E is

P_exp(E) = n(E) / N

Key ideas

• Experimental probability is based on actual data (observed frequency), not on theoretical reasoning about equally likely outcomes.
• Relative frequency = number of favourable outcomes observed ÷ total trials.
• As the number of trials increases, the experimental probability tends to approach the theoretical probability (Law of Large Numbers).
• Limitations: results depend on the number of trials and how the experiment is conducted (sampling bias, measurement errors).

How to do an experiment

1. Define the event E clearly (e.g., getting a head when tossing a coin).
2. Decide number of trials N and perform the experiment N times.
3. Count n(E) = number of times E occurred.
4. Compute P_exp(E) = n(E)/N. Optionally convert to decimal or percent.
5. Compare with theoretical probability (if known) and comment on differences due to randomness and sample size.

Law of Large Numbers (informal)

When an experiment is repeated many times, the experimental probability of an event tends to get closer to its theoretical probability. Small samples can show large fluctuations; larger samples give more stable estimates.

What is being counted? In probability we count the possible outcomes of a random experiment. The set of all possible outcomes is called the sample space (denoted S). An event is a subset of S. If all outcomes in S are equally likely, probability of an event E is P(E) = n(E) / n(S), where n(X) denotes the number of elements of set X.

Why list outcomes? Listing outcomes helps ensure you have found the complete sample space and that no outcome is counted twice. For small experiments it gives a direct way to compute probabilities by counting.

Common listing techniques

• Complete listing (enumeration): Write every outcome explicitly (useful when S is small).
• Tree diagram: Draw branches for sequential choices. Each path from root to leaf represents one outcome. Count leaves to get n(S).
• Tabular (matrix) method: Use a table/grid for two-step experiments (e.g. two dice → 6×6 grid = 36 outcomes).
• Systematic listing: Use a fixed order (alphabetical or numeric) to avoid duplicates and omissions.
• Fundamental principle of counting (multiplication rule): If an experiment has stages with a choices in stage 1, b choices in stage 2, ..., then total outcomes = a × b × ... (when choices are independent and order matters). This replaces explicit listing when numbers are larger.

Important notes:

• Decide whether order matters (e.g., arrangement AB is different from BA) and whether repetition is allowed. These affect counting.
• Always check that listed outcomes are equally likely before using simple ratio formula for probability.

What is Probability? Probability measures how likely an event is to occur. For a well-defined experiment with a sample space S (all possible equally likely outcomes) and an event E (a subset of S), the classical probability is P(E) = n(E) / n(S), where n(.) denotes number of outcomes.

Key ideas:

• Sample space (S): all possible outcomes of an experiment (e.g., for a fair die S = {1,2,3,4,5,6}).
• Equally likely outcomes: classical formula assumes outcomes have equal chance.
• Experimental (empirical) probability: found by repeating the experiment many times: P(E) ≈ (frequency of E)/(total trials).
• Complementary event: E' (not E). P(E') = 1 − P(E).
• Mutually exclusive events: A and B cannot happen together. Then P(A ∪ B) = P(A) + P(B).

Standard problem types you will meet in Class 9:

1. Single simple event with equally likely outcomes (coin, die, spinner).
2. Drawing objects from a bag — count favorable and total outcomes.
3. Events using complements (e.g., “at least one” problems solved by 1 − P(none)).
4. Combined events when outcomes are listed as ordered pairs (two tosses, two dice) — build sample space as Cartesian product.
5. Experimental probability tasks — perform trials, record frequencies, compare with theoretical values.
6. Mutually exclusive events and simple additions of probabilities.

Problem-solving steps (concise):

1. Define the experiment and identify S.
2. List/count n(S) and n(E) (favourable outcomes).
3. Check if outcomes are equally likely; if yes, use P(E)=n(E)/n(S).
4. Use complements or combinations if direct counting is hard.
5. Give final answer as a fraction, decimal or percentage, simplified.

What the topic covers: Problem-solving strategies in probability teach systematic ways to model random experiments, count outcomes, and compute probabilities accurately using basic rules (equally likely outcomes, complement, addition for disjoint events, multiplication for sequential events).

Step-by-step strategy (use this checklist for each problem)

• 1. Read and identify the experiment. What is being done (tossing, drawing, rolling)? How many trials?
• 2. Define the sample space S. List all possible outcomes explicitly if small, or describe them (e.g., all ordered pairs for two dice).
• 3. Check the equally-likely assumption. If outcomes are equally likely, P(E)=n(E)/n(S). If not, consider weighting or empirical frequency.
• 4. Count favourable outcomes n(E). Use listing, symmetry, multiplication principle, combinations, or complementary counting to simplify.
• 5. Use rules to compute probability. Apply P(E)=n(E)/n(S), or use complement, addition or multiplication rules as appropriate.
• 6. Draw a diagram if needed. Tree diagrams for sequential experiments, Venn diagrams for unions/intersections, tables for two-dimensional sample spaces, area models for geometric probability.
• 7. Check the answer. Probabilities must be between 0 and 1; reasonableness check using symmetry or complementary probabilities often helps.

Common short-cuts and tips

• Use complement: When "at least one" or "not" problems are present, it is often easier to compute P(E') and subtract from 1.
• Exploit symmetry: Symmetric outcomes (e.g., fair coin or fair die) let you count one type then multiply by symmetry factor.
• Use tree diagrams: For sequential events (with or without replacement), trees show all ordered outcomes and probabilities clearly.
• Area/proportional models: For continuous or geometric problems, represent outcomes by areas; probability = shaded area / total area.

When to use which diagram

• Tree diagram — sequential experiments (ordered outcomes, dependent or independent draws).
• Venn diagram — events involving union/intersection/complement relationships.
• Matrix/table (grid) — two independent discrete variables (e.g., two dice).
• Area model — geometric probability (points chosen at random inside a region).

Following these organized steps prevents common mistakes (forgetting order, double-counting, or assuming equal likelihood incorrectly) and makes even non-trivial problems manageable.