Chapter 15 — Probability

Random experiment: An action or process that can be repeated under identical conditions and has one or more possible results (outcomes), where the result cannot be predicted with certainty beforehand. Examples: tossing a coin, rolling a die, drawing a card.

Outcome (sample point): A single possible result of a random experiment. The set of all possible outcomes is called the sample space (denoted S). Each outcome in S is an elementary event.

Sample space (S): The collection of all possible outcomes. It can be represented as a list or set. Example: For a fair die S = {1,2,3,4,5,6}; for a coin S = {H, T}.

Equally likely outcomes: Outcomes are equally likely if each has the same chance of occurring (common in symmetric experiments like fair coins, fair dice).

Event: Any subset of the sample space (one or more outcomes). A simple (elementary) event contains a single outcome; a compound event contains several.

Relation to probability (brief): When outcomes are equally likely, the probability of event E is the ratio of the number of favourable outcomes to the total number of outcomes: P(E) = n(E) / n(S). (See formulas below.)

Deterministic vs random: Deterministic experiments always give the same outcome under identical conditions (no randomness). Random experiments may give different outcomes when repeated.

How to list S: Use outcome listing, tree diagrams (for sequences), or systematic counting rules (product rule) to build S. For combined experiments with m outcomes in the first and n in the second, total outcomes = m × n.

Definition: In a random experiment, the sample space (denoted by S or Ω) is the set of all possible outcomes. Each element of S is called a sample point. For example, when tossing one fair coin, S = {H, T} where H = head and T = tail.

Why it matters: The sample space is the foundation for defining events and computing probabilities. An event is any subset of S. To find the probability of an event, you must first know what outcomes exist (S) and which of them make the event happen.

Types of sample spaces:

• Finite (discrete): number of outcomes is finite (e.g., tossing a coin, rolling a die).
• Countably infinite: outcomes can be listed in sequence (e.g., number of tosses until first head).
• Uncountable/continuous: outcomes form a continuum (e.g., all possible arrival times between 0 and 60 minutes). In Class 10 most problems use finite or countable sample spaces.

How to list a sample space: Use set notation with commas and braces. For compound experiments, outcomes are ordered pairs (or tuples). Example: tossing two coins, S = {(H,H), (H,T), (T,H), (T,T)}. For rolling two dice, S is the set of 36 ordered pairs (1,1),(1,2),...,(6,6).

Important remarks:

• Order matters when outcomes are ordered pairs. If order does not matter, states should be listed appropriately (e.g., drawing two balls without order).
• When outcomes are equally likely, probability of an event E is n(E)/n(S), where n(X) denotes number of elements of set X.

Examples in words: "Choose one card from a pack" → S = {52 playing cards}; "Observe today’s weather" → S = {Sunny, Cloudy, Rainy, ...}; "Measure lifetime of a bulb" → continuous sample space (positive real numbers).

Definition: In probability, an event is any subset of the sample space (S) of a random experiment. An event E contains one or more outcomes (elements) of S. If S is the set of all possible outcomes, then E ⊆ S.

Types of events:

• Simple (elementary) event: contains exactly one outcome (e.g., rolling a 4).
• Compound event: contains more than one outcome (e.g., rolling an even number).
• Sure (certain) event: E = S; P(E) = 1.
• Impossible event: E = ∅; P(E) = 0.
• Complementary event: E^c (or E') is the set of outcomes in S not in E; P(E^c) = 1 − P(E).
• Mutually exclusive (disjoint) events: A and B are mutually exclusive if A ∩ B = ∅ (they cannot occur together).
• Independent events: occurrence of one does not affect the probability of the other; P(A ∩ B) = P(A)P(B).

How probability of an event is computed (equally likely outcomes): If all outcomes in S are equally likely, P(E) = n(E) / n(S), where n(E) is number of favourable outcomes and n(S) is total number of outcomes.

Alternative (empirical) approach: For experiments repeated many times, P(E) ≈ (number of times E occurred) / (total trials).

Basic properties: P(∅)=0, P(S)=1, 0 ≤ P(E) ≤ 1. For any events A and B: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive: P(A ∪ B) = P(A) + P(B). For complement: P(E^c) = 1 − P(E).

Basic terms: A random experiment is an action with uncertain outcomes. The sample space S is the set of all possible outcomes. An event A is any subset of S (one or more outcomes).

Types of events:

• Sure (certain) event: an event that always occurs: A = S. P(A) = 1.
• Impossible event: an event that cannot occur: A = Ø. P(A) = 0.
• Elementary (simple) event: contains exactly one outcome (e.g., getting 5 on a die).
• Compound event: contains two or more outcomes (e.g., getting an even number on a die: {2,4,6}).
• Equally likely events: outcomes with the same probability (e.g., fair coin faces, fair die faces).
• Mutually exclusive (disjoint) events: cannot occur together: A ∩ B = Ø. Example: getting 2 and getting 5 on a single die roll.
• Exhaustive events: a collection of events whose union is the whole sample space (they cover all outcomes).
• Complementary events: A and A' (not A) are complements: A ∪ A' = S and A ∩ A' = Ø. P(A') = 1 − P(A).
• Independent events: occurrence of one does not affect the probability of the other. For A and B independent: P(A ∩ B) = P(A)·P(B).

Relations and important rules:

• Addition (general): P(A ∪ B) = P(A) + P(B) − P(A ∩ B). This avoids double counting the intersection.
• Addition (mutually exclusive): If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
• Multiplication (independent): If A and B are independent, P(A ∩ B) = P(A)P(B).
• Conditional probability: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. If P(A|B) = P(A) then A and B are independent.
• Complement rule: P(A') = 1 − P(A).
• De Morgan’s laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.
• Classical probability (for equally likely outcomes): If n(S) is finite and equally likely, P(A) = n(A) / n(S).

Classical definition: If an experiment has a finite sample space S in which all outcomes are equally likely, and E is an event (a set of favourable outcomes), then the probability of E is

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

Here n(E) is the number of favourable outcomes and n(S) is the total number of possible outcomes. The classical definition applies only when all individual outcomes have the same chance of occurring (equally likely).

How to apply (steps):

1. Describe the sample space S and count n(S).
2. Identify the favourable outcomes that make up event E and count n(E).
3. Compute P(E) = n(E)/n(S). Simplify the fraction.

Properties (direct consequences):

• P(E) is between 0 and 1: 0 ≤ P(E) ≤ 1.
• Probability of the certain event (sample space): P(S) = 1.
• Probability of the impossible event (empty set): P(∅) = 0.
• Complement rule: P(E') = 1 − P(E), where E' is the complement of E in S.
• Addition for mutually exclusive events A and B: P(A ∪ B) = P(A) + P(B) (if A ∩ B = ∅).

When classical definition is not valid: If outcomes are not equally likely (for example, a biased coin, or selecting days weighted by activity), the classical formula n(E)/n(S) is not valid and a different approach (relative frequency, axiomatic) is needed.

Definition and setup: For a random experiment, the sample space S is the set of all possible outcomes. An event is any subset E of S. The probability P(E) is a number assigned to E that measures how likely E is to occur. In the classical (equally likely) case, P(E) = (number of favourable outcomes)/(total number of outcomes).

Interpretations: Two common interpretations are the classical (combinatorial) interpretation—useful when outcomes are equally likely—and the relative frequency (empirical) interpretation—P(E) ≈ (frequency of E)/N for large N repeats of the experiment.

Basic properties (with brief reasoning):

• Non-negativity: 0 ≤ P(E) for every event E. (Probabilities cannot be negative.)
• Normalization: P(S) = 1. The probability that some outcome in the sample space occurs is certain.
• Null event: P(∅) = 0. The empty event (no outcome) has probability zero.
• Complement rule: P(E') = 1 − P(E), where E' (or E^c) is the complement of E in S. (Either E occurs or it does not.)
• Addition rule (two events): For any two events A and B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). This avoids double-counting outcomes that are in both A and B.
• Mutually exclusive events: If A and B are mutually exclusive (A ∩ B = ∅), then P(A ∪ B) = P(A) + P(B).
• Monotonicity: If A ⊆ B then P(A) ≤ P(B). (A smaller event cannot be more likely than a larger one containing it.)
• Difference rule: P(B \ A) = P(B) − P(A ∩ B). (Probability of outcomes in B but not in A.)

Remarks: These properties follow directly from the set relationships among events and from the basic idea that probability distributes over disjoint unions. For Class 10 problems you usually apply the classical formula when outcomes are equally likely and then use the properties above to combine events.

Overview: Probability measures how likely an event is to occur. There are three common methods to compute probability taught in Class 10: the classical (theoretical) method, the experimental (empirical or relative-frequency) method, and the axiomatic approach. In many problems you also use counting techniques (combinations/permutations), complementary rules, and addition/multiplication rules.

1. Classical (Theoretical) Method

Use when all outcomes of a random experiment are equally likely. If S is the sample space with n(S) equally likely outcomes and E is an event with n(E) favourable outcomes,

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

Counting n(E) and n(S) often uses combinations/permutations (e.g., number of ways to choose or arrange objects).

2. Experimental (Empirical or Relative-frequency) Method

Use when you perform actual trials and record results. If an experiment is repeated N times and event E occurs f(E) times, the experimental probability is:

Experimental P(E) ≈ f(E)/N

As N becomes large, f(E)/N tends to the true probability (law of large numbers).

3. Axiomatic Approach

Probability is defined on a sample space S with a function P satisfying the axioms: (i) 0 ≤ P(E) ≤ 1, (ii) P(S) = 1, (iii) For disjoint events A and B, P(A ∪ B) = P(A) + P(B). From these axioms several useful rules follow (complement, addition, conditional, multiplication rules).

Useful rules and techniques

• Complement rule: P(E') = 1 − P(E) (useful for 'at least one' problems).
• Addition rule: For any events A and B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B).
• Multiplication and conditional: P(A ∩ B) = P(A) · P(B|A). If A and B are independent, P(A ∩ B) = P(A)·P(B).
• Counting: Use combinations C(n, r) and permutations P(n, r) to count outcomes when items are chosen/arranged.

When to use which method: Use the classical method for well-defined, equally likely outcomes (coins, fair dice, cards). Use experimental method when you actually perform trials or when theoretical model is unavailable. Use axiomatic approach for rigorous properties and when combining events.

What probability is: Probability measures how likely an event is to occur. For equally likely outcomes, probability of an event E is P(E) = n(E) / n(S), where n(E) is number of favourable outcomes and n(S) is total number of outcomes in the sample space S.

Basic terms: sample space (all possible outcomes), event (a subset of outcomes), complement (E' = outcomes where E does not happen), mutually exclusive events (cannot occur together), independent events (one does not affect the other).

Standard examples: many class‑10 problems use a small finite sample space with equally likely outcomes: coin tosses, throws of a die, picking cards from a 52‑card deck, selecting numbered tickets, and drawing balls from an urn. These examples teach counting favourable outcomes, using complements and simple product/addition rules.

Common techniques:

• List the sample space explicitly for small experiments (e.g., for two coin tosses S = {HH, HT, TH, TT}).
• Use counting (combinations/permutations) when sample space is larger (e.g., choosing 2 from n without order).
• Use the complement rule P(E) = 1 − P(E') when it is easier to count E'.
• Use tree diagrams for sequential experiments (with or without replacement).

Applications in real life: predicting weather (chance of rain), quality control (probability of defective item), games of chance (cards, dice), insurance and risk assessment (estimating loss probabilities), genetics (probability of inheriting traits), and reliability of systems (probability of component failure).

How to approach CBSE problems: identify whether outcomes are equally likely, write S and E, count n(S) and n(E), simplify the fraction. For sequential draws check if replacement makes trials independent; without replacement use conditional counting (reduce totals after each draw).

Problem‑solving strategies in probability are systematic ways to translate a real situation into mathematical statements and compute probabilities correctly. The basic classical idea used in Class 10 is: Probability of an event E = number of favourable outcomes / number of equally likely outcomes (P(E) = n(E)/n(S)).

Follow these stepwise strategies for most problems:

1. Read and understand: Identify the random experiment, what one trial is, and what the required event E means in plain words.
2. Define the sample space S: List all equally likely outcomes (coin tosses, faces of a die, balls in a bag, ordered pairs for two experiments, etc.). Use a table, tree diagram, or grid when helpful.
3. Identify favourable outcomes: Write the outcomes that satisfy the event. Count them carefully; watch for order (ordered vs unordered), replacement vs no replacement.
4. Use appropriate counting tools: Use the basic counting principle (product rule), simple permutations/combinations if needed, or complementary counting to simplify counting of favourable outcomes.
5. Apply probability formulas: Compute P(E)=n(E)/n(S). Use complement rule P(E') = 1 − P(E), addition rule for unions, and multiplication rule for independent events where appropriate.
6. Check edge cases and reasonableness: Probability should be between 0 and 1. If result seems odd, try solving with complementary or direct counting again.

Common techniques/shortcuts:

• Complement rule: If counting E is hard, count E' (the complement) and use P(E)=1−P(E').
• Symmetry: If outcomes are symmetric (e.g., fair coin, fair die), exploit symmetry to reduce counting.
• Tree diagrams: Useful for sequential experiments (with/without replacement). They visually encode outcome probabilities or counts.
• Tables and grids: Use a 6x6 grid for two dice, or a matrix for two-stage experiments, to list ordered pairs.
• Venn diagrams: Helpful when dealing with unions, intersections and complements of events.

Keep clear whether order matters and whether trials are independent. In "without replacement" situations the sample space changes after each draw; then count ordered outcomes or use conditional probabilities.

Worked mini example (conceptual): A bag has 3 red and 2 blue balls. One ball is drawn at random. S = {R,R,R,B,B} (5 equally likely outcomes). Probability of red = 3/5. If two balls are drawn without replacement and order matters, S has 5x4 = 20 ordered outcomes; count favourable ordered pairs or use complementary/conditional reasoning.

Use these strategies together: careful modeling, choosing counting method, using complement/symmetry, and then applying the classical formula. This approach will solve most Class 10 probability problems reliably.