'Probability'

Introduction to Probability

Probability is a branch of mathematics that deals with quantifying uncertainty. It originated from gambling problems studied by mathematicians like Pascal, Fermat, and Huygens in the 17th century.

Random Experiment

A random experiment is an experiment whose outcome cannot be predicted in advance but all possible outcomes are known.

Conditions for a Random Experiment

1. All possible outcomes are known in advance.
2. The experiment can be repeated under identical conditions.
3. The outcome of any particular trial is not predictable.

Examples

• Tossing a coin (outcomes: H, T)
• Rolling a die (outcomes: 1, 2, 3, 4, 5, 6)
• Drawing a card from a deck (52 possible outcomes)

Sample Space

The set of all possible outcomes of a random experiment is called the sample space, denoted by S.

Each outcome is called a sample point.

Common Sample Spaces

• Coin toss: S = {H, T}
• Two coins tossed: S = {HH, HT, TH, TT}
• Die rolled: S = {1, 2, 3, 4, 5, 6}
• Two dice rolled: S has 36 ordered pairs.

Events

An event is a subset of the sample space. If the outcome of the experiment belongs to the event, the event is said to have occurred.

Types of Events

Simple (Elementary) Event: An event with a single sample point. Example: Getting a 6 on a die. E = {6}.

Compound Event: An event with more than one sample point. Example: Getting an even number on a die. E = {2, 4, 6}.

Impossible Event: An event with no sample points. Denoted by phi. Example: Getting a 7 on a die.

Sure (Certain) Event: An event equal to the entire sample space S. Example: Getting a number less than 7 on a die.

Complementary Event: The complement of event A is A' = S - A. P(A') = 1 - P(A).

Mutually Exclusive Events: Two events A and B are mutually exclusive if they cannot occur simultaneously. A cap B = phi.

Exhaustive Events: Events E_1, E_2, ..., E_n are exhaustive if their union equals the sample space S.

Classical Definition of Probability

If a random experiment has n equally likely outcomes, and an event A has m favourable outcomes, then:

P(A) = m/n = text(Number of favourable outcomes)/text(Total number of possible outcomes)

Axioms of Probability

1. 0 <= P(A) <= 1 for any event A.
2. P(S) = 1 (sure event).
3. If A and B are mutually exclusive, P(A cup B) = P(A) + P(B).

Addition Theorem of Probability

For Two Events

P(A cup B) = P(A) + P(B) - P(A cap B)

For Mutually Exclusive Events

If A cap B = phi, then P(A cup B) = P(A) + P(B).

For Three Events

P(A cup B cup C) = P(A) + P(B) + P(C) - P(A cap B) - P(B cap C) - P(C cap A) + P(A cap B cap C)

Odds in Favour and Against

If m outcomes favour event A and n outcomes do not favour A (total = m+n):

Odds in favour of A: m : n Odds against A: n : m Probability of A: m/(m+n)

Worked Examples

Example 1: A coin is tossed twice. Find the probability of getting at least one head. Solution: Sample space S = {HH, HT, TH, TT}. n(S) = 4. Event A = at least one head = {HH, HT, TH}. n(A) = 3. P(A) = 3/4.

Example 2: Two dice are rolled. Find the probability of getting a sum of 7. Solution: Total outcomes = 6 x 6 = 36. Favourable outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes. P = 6/36 = 1/6.

Example 3: A card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting a king or a heart. Solution: P(King) = 4/52, P(Heart) = 13/52, P(King of Hearts) = 1/52. By addition theorem: P = 4/52 + 13/52 - 1/52 = 16/52 = 4/13.

Example 4: Find the probability that a leap year has 53 Sundays. Solution: A leap year has 366 days = 52 weeks + 2 days. The remaining 2 days can be: (Sun, Mon), (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat), (Sat, Sun). Favourable: (Sun, Mon), (Sat, Sun) = 2 outcomes. P = 2/7.

Common Mistakes

1. Equally likely assumption: Verify that outcomes are equally likely before using classical definition.
2. Addition theorem sign: For non-mutually exclusive events, subtract the intersection.
3. 'At least one' problems: Often P(at least one) = 1 - P(none) is easier.
4. Deck of cards: 52 cards, 4 suits of 13 cards each. Know the composition before solving.

ISC Exam Focus

• Theory (70%): Definitions, sample space construction, types of events, axioms.
• Application (30%): Probability calculations using classical definition, addition theorem.
• ISC frequently asks: "Find the probability of ... when two dice are rolled."
• 4-6 mark questions on event types and compound probability.

Self-Test Questions

Q1: A coin is tossed three times. Write the sample space. Answer: S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. 8 equally likely outcomes.

Q2: Find the probability of getting a sum of at least 10 when two dice are rolled. Answer: Favourable: (4,6), (5,5), (5,6), (6,4), (6,5), (6,6) = 6. Total = 36. P = 6/36 = 1/6.

Q3: A card is drawn from a deck. Find the probability of it being a red king. Answer: Red kings: King of Hearts, King of Diamonds = 2. P = 2/52 = 1/26.

Q4: In a single throw of two dice, find the probability of getting a doublet (same number on both). Answer: Favourable: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) = 6. P = 6/36 = 1/6.

Q5: If P(A) = 0.5, P(B) = 0.4, and P(A cap B) = 0.2, find P(A cup B). Answer: P(A cup B) = 0.5 + 0.4 - 0.2 = 0.7.

Q6: Three coins are tossed. Find the probability of getting exactly two heads. Answer: Favourable: HHT, HTH, THH = 3. Total = 8. P = 3/8.