Probability (Class 10): Notes और MCQ Practice

आसान भाषा में

Probability का सीधा मतलब है किसी event के होने का chance या संभावना। Class 10 में हम Theoretical Probability पढ़ते हैं, जहाँ हम बिना actual experiment किए, सभी possible outcomes को equally likely मानकर गणित के ज़रिए संभावना निकालते हैं।

Important Formulas & Key Points

• Basic Formula: P(E) = (Number of favourable outcomes) ÷ (Total number of possible outcomes)
• Range: 0 ≤ P(E) ≤ 1 (Probability कभी negative या 1 से ज़्यादा नहीं हो सकती)।
• Sure Event: जिसकी संभावना पक्की हो, P(E) = 1.
• Impossible Event: जो कभी हो ही न सके, P(E) = 0.
• Complementary Event: P(E) + P(not E) = 1, यानी P(not E) = 1 − P(E).
• Deck of 52 Cards:
• 26 Red (13 Hearts, 13 Diamonds) और 26 Black (13 Spades, 13 Clubs).
• कुल 12 Face Cards होते हैं (4 Kings, 4 Queens, 4 Jacks).

2 Memory Tricks

1. "0 से 1 की जेल": Probability हमेशा 0 और 1 की सीमा में बंद रहती है। अगर board exam में आपका answer negative या 1 से बड़ा (जैसे 1.5 या 5/4) आए, तो तुरंत समझ जाएं कि calculation गलत है!
2. "J-Q-K का चेहरा": सिर्फ Jack, Queen, और King पर ही चेहरा बना होता है (3 × 4 = 12 Face Cards)। Ace (इक्का) Face card नहीं होता!

2 Common Board-Exam Mistakes

1. Ace को Face card मान लेना: कई बच्चे Ace को भी Face card जोड़कर कुल 16 लिख देते हैं, जिससे marks कट जाते हैं। Face cards सिर्फ 12 होते हैं।
2. "At least" और "At most" में गलती: 'At least 1 head' का मतलब है कम-से-कम 1 (1 या 1 से ज़्यादा), जबकि 'At most 1 head' का मतलब है ज़्यादा-से-ज़्यादा 1 (0 या 1)।

Probability - Class 10 NCERT Study Notes

1. Chapter in Short

In Class 9, you learned about experimental probability based on the actual results of experiments. In Class 10, the focus shifts to theoretical (classical) probability.

Theoretical probability helps us calculate the chance of an event happening before actually performing the experiment, assuming all outcomes are equally likely. It deals with everyday situations such as tossing coins, rolling dice, picking cards from a deck, or selecting colored balls from a bag.

2. Key Concepts Explained Simply

1. Equally Likely Outcomes

Outcomes of an experiment are called equally likely if each outcome has the same chance of happening as any other.

• Example: When you flip an unbiased coin, getting a Head (H) and getting a Tail (T) are equally likely. Neither is favored.

2. Elementary Event

An event having only one outcome of the experiment is called an elementary event.

• Example: Rolling a die once and getting a 4 is an elementary event.
• Crucial Rule: The sum of the probabilities of all the elementary events of an experiment is always equal to 1.

3. Complementary Events

For any event E, the event "not E" (often written as E̅ or E') is called the complement of E.

• Example: If event E is "raining today", then not E is "not raining today".
• Formula: P(E) + P(not E) = 1, which means P(not E) = 1 − P(E).

4. Impossible and Sure Events

• Impossible Event: An event that cannot happen at all. Its probability is 0.
• Example: Rolling a 7 on a standard 6-sided die. P(getting 7) = 0 ÷ 6 = 0.
• Sure (or Certain) Event: An event that is guaranteed to happen. Its probability is 1.
• Example: Rolling a number less than 7 on a standard 6-sided die. P(number < 7) = 6 ÷ 6 = 1.

5. Range of Probability

The probability of any event always lies between 0 and 1 (inclusive).

• 0 ≤ P(E) ≤ 1
• Probability can never be negative (e.g., −0.5 is impossible).
• Probability can never be greater than 1 (e.g., 1.5 or 120% is impossible).

6. Standard Card Deck Breakdown (52 Cards)

A standard pack of playing cards has 52 cards in total:

• 2 Colors: Red (26 cards) and Black (26 cards).
• 4 Suits (13 cards each):
• Red suits: Hearts (13), Diamonds (13)
• Black suits: Spades (13), Clubs (13)
• Cards in each suit: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack (J), Queen (Q), King (K).
• Face Cards: Jacks, Queens, and Kings have faces on them.
• Total face cards = 4 Jacks + 4 Queens + 4 Kings = 12 face cards (6 red and 6 black).
• Note: An Ace is not considered a face card.

3. Important Definitions, Formulas & Facts

• Classical Probability Formula:

P(E) = (Number of outcomes favourable to E) ÷ (Number of all possible outcomes of the experiment)

• Probability Limits:

0 ≤ P(E) ≤ 1

• Sum of Elementary Probabilities:

P(E₁) + P(E₂) + P(E₃) + ... + P(Eₙ) = 1

• Complementary Rule:

P(E) + P(E̅) = 1 => P(E̅) = 1 − P(E)

• Tossing Coins:
• 1 coin: Total outcomes = 2 {H, T}
• 2 coins: Total outcomes = 4 {HH, HT, TH, TT}
• 3 coins: Total outcomes = 8 {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

• Rolling Dice:
• 1 die: Total outcomes = 6 {1, 2, 3, 4, 5, 6}
• 2 dice: Total outcomes = 6 × 6 = 36 outcomes {(1, 1), (1, 2), ..., (6, 6)}

4. Solved Examples

Example 1: Complementary Events

Question: If P(E) = 0.05, what is the probability of 'not E'?

Solution:

• We know that P(E) + P(not E) = 1
• Therefore, P(not E) = 1 − P(E)
• P(not E) = 1 − 0.05 = 0.95

Answer: 0.95

Example 2: Playing Cards

Question: One card is drawn from a well-shuffled deck of 52 cards. Calculate the probability that the card will be:

(i) a king of red colour

(ii) a face card

(iii) a spade

Solution:

Total number of possible outcomes = 52

• (i) A king of red colour:

There are 2 red kings (King of Hearts and King of Diamonds).

Favourable outcomes = 2

P(red king) = 2 ÷ 52 = 1 ÷ 26

• (ii) A face card:

There are 12 face cards (4 Jacks, 4 Queens, 4 Kings).

Favourable outcomes = 12

P(face card) = 12 ÷ 52 = 3 ÷ 13

• (iii) A spade:

There are 13 spades in a deck.

Favourable outcomes = 13

P(spade) = 13 ÷ 52 = 1 ÷ 4

Example 3: Tossing Two Coins

Question: Two coins are tossed simultaneously. Find the probability of getting:

(i) at least one head

(ii) at most one head

Solution:

All possible outcomes when 2 coins are tossed:

S = {HH, HT, TH, TT}

Total number of possible outcomes = 4

• (i) At least one head:

"At least one" means 1 or more heads.

Favourable outcomes = {HT, TH, HH} (3 outcomes)

P(at least one head) = 3 ÷ 4

• (ii) At most one head:

"At most one" means 1 or fewer heads (0 or 1 head).

Favourable outcomes = {TT, HT, TH} (3 outcomes)

P(at most one head) = 3 ÷ 4

Example 4: Two Dice Rolled Together

Question: Two dice are thrown at the same time. What is the probability that the sum of the two numbers appearing on top of the dice is:

(i) 8

(ii) 13

(iii) less than or equal to 12

Solution:

Total number of outcomes = 6 × 6 = 36

• (i) Sum is 8:

Favourable outcomes = (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)

Number of favourable outcomes = 5

P(sum is 8) = 5 ÷ 36

• (ii) Sum is 13:

The maximum possible sum is 6 + 6 = 12. Sum of 13 cannot happen.

Number of favourable outcomes = 0

P(sum is 13) = 0 ÷ 36 = 0 (Impossible event)

• (iii) Sum is less than or equal to 12:

Every possible pair has a sum between 2 and 12.

Number of favourable outcomes = 36

P(sum ≤ 12) = 36 ÷ 36 = 1 (Sure event)

5. Common Mistakes & Quick Revision Points

Common Mistakes to Avoid:

1. "At least" vs "At most":

• At least 1 means 1 or more (minimum 1).
• At most 1 means 1 or fewer (maximum 1, including 0).

1. Confusing Aces with Face Cards:

• Aces do not have faces. There are only 12 face cards (Jacks, Queens, Kings).

1. Writing Probabilities Greater than 1 or Less than 0:

• If your answer is negative or bigger than 1, you made a calculation error.

1. Order in Dice and Coins:

• In two coins, (H, T) and (T, H) are two separate outcomes, not one.
• In two dice, (2, 5) and (5, 2) are distinct outcomes.

Quick Revision Points:

• Formula: P(E) = Favourable outcomes ÷ Total outcomes
• Range: 0 ≤ P(E) ≤ 1
• Impossible event has P = 0; Sure event has P = 1
• Sum of all elementary events is 1
• Complementary events: P(E) + P(not E) = 1