In-Depth: Probability
CLAT Application & Relevance
Importance: Very Low. While Probability is a standard quantitative topic, direct and complex probability questions are highly infrequent in CLAT QT. If tested, it's usually very basic (e.g., probability of a simple event with coins, dice, or cards) within a larger numerical reasoning context. The focus remains on interpreting data rather than calculating chances of complex events.
How it's tested: Rarely, as a simple direct question on the likelihood of an outcome in a given scenario within a caselet. Basic understanding of probability concepts is sufficient.
Section 1: Core Concepts & Formulas
Probability is a measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
Key Definitions
- Experiment: A process that produces a well-defined outcome. (e.g., tossing a coin, rolling a die).
- Outcome: A possible result of an experiment.
- Sample Space (S): The set of all possible outcomes of an experiment.
- Event (E): A subset of the sample space; a collection of specific outcomes.
- Mutually Exclusive Events: Events that cannot occur at the same time (e.g., getting a Head and a Tail on a single coin toss).
- Independent Events: Events where the outcome of one does not affect the outcome of the other (e.g., two successive coin tosses).
Fundamental Formulas
- Basic Probability of an Event E:
P(E) = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes) - Probability of an event NOT happening:
P(not E) = 1 - P(E) - Addition Rule (for Mutually Exclusive Events):
P(A or B) = P(A) + P(B) - Addition Rule (for Non-Mutually Exclusive Events):
P(A or B) = P(A) + P(B) - P(A and B) - Multiplication Rule (for Independent Events):
P(A and B) = P(A) * P(B)
Section 2: Solved CLAT-Style Examples (Basic Application)
Example 1: Probability with Dice (Basic)
Passage Context: "In a legal strategy game, players roll a standard six-sided die. A special bonus is awarded if a player rolls a number greater than 4."
Question: "What is the probability of rolling a number greater than 4?"
Detailed Solution:
1. Identify Sample Space (Total Outcomes): For a standard die, S = {1, 2, 3, 4, 5, 6}. Total possible outcomes = 6.
2. Identify Favorable Outcomes (Event E): Numbers greater than 4 are {5, 6}. Number of favorable outcomes = 2.
3. Apply Probability Formula: P(E) = (Favorable Outcomes) / (Total Outcomes)
P(>4) = 2 / 6 = 1/3.
Answer: The probability of rolling a number greater than 4 is 1/3.
Example 2: Probability with Cards (Basic)
Passage Context: "A lawyer is playing a card game based on a well-shuffled standard deck of 52 playing cards. They need to draw a specific type of card to win a round."
Question: "What is the probability of drawing a King or a Queen from a standard deck in a single draw?"
Detailed Solution:
1. Identify Total Outcomes: Total cards in a deck = 52.
2. Identify Favorable Outcomes:
- Number of Kings = 4 (King of Spades, Clubs, Hearts, Diamonds)
- Number of Queens = 4 (Queen of Spades, Clubs, Hearts, Diamonds)
- Are these mutually exclusive? Yes, a card cannot be both a King and a Queen.
3. Calculate Probability of drawing a King: P(King) = 4/52 = 1/13.
4. Calculate Probability of drawing a Queen: P(Queen) = 4/52 = 1/13.
5. Apply Addition Rule for Mutually Exclusive Events: P(King or Queen) = P(King) + P(Queen)
= 1/13 + 1/13 = 2/13.
Answer: The probability of drawing a King or a Queen is 2/13.