CLASS X : CHAPTER - 15 PROBABILITY

CLASS X : CHAPTER - 15 PROBABILITY

IMPORTANT FORMULAS & CONCEPTS

Probability Definitions

• Experimental/Empirical Probability P(E):
  \( P(E) = \frac{\text{Number of trials in which the event happened}}{\text{Total number of trials}} \)
• Theoretical/Classical Probability P(A):
  \( P(A) = \frac{\text{Number of outcomes favourable to A}}{\text{Number of all possible outcomes of the experiment}} \)

Key Event Types and Properties

• Mutually Exclusive Events: Two or more events where the occurrence of one prevents the occurrence of all others.
• Equally Likely Events: Events that have an equal chance of occurring.
• Elementary Event: An event having only one outcome. The sum of probabilities of all elementary events of an experiment is 1.
• Complementary Events: The event 'not E', denoted by \( \bar{E} \), is the complement of event E.
  \( P(E) + P(\bar{E}) = 1 \) or \( P(\bar{E}) = 1 - P(E) \)
• Sure/Certain Event: An event which is sure to occur. Its probability is 1.
• Impossible Event: An event which cannot occur. Its probability is 0.
• Range of Probability: For any event E, \( 0 \le P(E) \le 1 \).

Deck of Cards (52 Cards)
Divided into 4 suits of 13 cards each:

• Black Suits: Spades (♠), Clubs (♣)
• Red Suits: Hearts (♥), Diamonds (♦)
• Cards in each suit: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.
• Face Cards: Kings, Queens, and Jacks.

MCQ WORKSHEETS OVERVIEW

Worksheet-I
Focuses on basic probability calculations involving:

• Drawing numbered marbles from a box.
• Flipping coins.
• Throwing a die (prime numbers).
• Drawing cards marked with numbers 1 to 25 (multiples, primes, divisibility).
• Drawing colored balls from a bag.

Worksheet-II
Focuses on:

• Marbles of specific colors (blue, white, red) and combinations (red or blue).
• Die throws (prime numbers, numbers between ranges, odd/even).

Worksheet-III
Focuses on:

• Marbles (red, white, green, blue).
• Cards numbered 1 to 50 (divisibility by 3, 4, 7; prime numbers).

Worksheet-IV
Focuses on empirical probability using frequency tables:

• Coin tosses (1000 times, 200 times).
• Die throws (200 times).
• Traffic statistics (vehicles passing).
• Blood group distribution.
• Defective items (bulbs, pens) mixed with good ones.
• Telephone directory digit frequency.

Worksheet-V
Focuses entirely on a standard deck of 52 cards:

• Probabilities of specific cards (red king, face card, black face card, ace, etc.).
• Compound conditions (face card or ace, king and red card).

PRACTICE QUESTIONS CATEGORIES

1. Dice Based Questions

• Single die outcomes (even, multiple of 3, prime, etc.).
• Two dice thrown simultaneously:
  • Sum of numbers (prime, even, multiple of 5, etc.).
  • Product of numbers.
  • Doublets (same number on both).
  • Difference of numbers.

2. Playing Cards Based Questions

• Drawing one card from a full deck.
• Scenarios where specific cards (e.g., face cards, spades) are removed before drawing.
• Probabilities involving specific subsets like only face cards or only diamonds.

3. Coin Based Questions

• Tossing two coins (at least one head, exactly two tails, etc.).
• Tossing three coins (at least 2 heads, exactly two tails, etc.).
• Tossing four coins (various combinations of heads and tails).

4. Bag/Balls Based Questions

• Drawing balls of different colors (red, black, white, green, blue).
• Probabilities of "not" occurring (e.g., not green).
• Compound events (white or green).
• Sequential drawing (drawing a ball, putting it back/not replacing).

5. Number Based Questions

• Drawing discs, cards, or tickets numbered in specific ranges (e.g., 1-90, 13-60).
• Conditions include: two-digit numbers, perfect squares, multiples, divisibility rules, prime numbers.
• Telephone number digit analysis.

6. Logical Reasoning & Area Based Questions

• General life scenarios: Rain probability, birthdays (same/different), defective items (pens, bulbs, shirts).
• Days in a year (leap year vs ordinary year probabilities for 53 Sundays).
• Lottery tickets and prizes.
• Geometric Probability: Helicopter crashing in a lake within a rectangular region, dropping a die into a circle within a rectangle.
• Spinning arrows (games of chance).