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Basic Probability Definition

Introduction

Probability यह मापती है कि कोई घटना होने की कितनी संभावना है। यह सभी higher-level probability concepts (जैसे conditional probability, Bayes’ theorem) की नींव बनती है। इस basic definition को समझने से coin toss, dice throw या card निकालने जैसी simple घटनाओं की likelihood आसानी से निकाली जा सकती है।

Pattern: Basic Probability Definition

Pattern

किसी event की probability = favourable outcomes की संख्या ÷ total possible outcomes.

Formula: P(E) = Favourable Outcomes / Total Outcomes

Probability के मान हमेशा 0 और 1 के बीच होते हैं।

Step-by-Step Example

Question

एक coin को एक बार toss किया जाता है। (i) Head आने की, (ii) Tail आने की probability निकालें।

Solution

  1. Step 1: सभी possible outcomes पहचानें

    Coin toss करने पर 2 possible outcomes होते हैं - Head (H) या Tail (T)।

  2. Step 2: Total outcomes गिनें

    Total outcomes = 2।

  3. Step 3: Favourable outcomes गिनें

    Head आने के favourable outcomes = 1 (सिर्फ H) ।

  4. Step 4: Probability compute करें

    इसलिए P(Head) = 1/2 और P(Tail) = 1/2।

  5. Final Answer:

    Head की probability = 1/2; Tail की probability = 1/2.

  6. Quick Check:

    Total probability = 1/2 + 1/2 = 1 ✅ (सही है, क्योंकि सभी outcomes की कुल probability = 1 होती है)

Quick Variations

1. Coin की जगह die toss करना (even number आने की probability)।

2. एक bag से colored ball निकालना (red आने की probability)।

3. Multiple coins या dice - combined outcomes की probability।

Trick to Always Use

  • Step 1: Total possible outcomes स्पष्ट रूप से पहचानें।
  • Step 2: केवल पूछे गए event के favourable outcomes गिनें।
  • Step 3: Formula P(E) = Favourable / Total लागू करें।

Summary

Summary

Basic Probability Definition pattern में:

  • Probability = Favourable / Total।
  • सभी probabilities 0 और 1 के बीच रहती हैं।
  • सभी possible outcomes की कुल probability = 1 होती है।
  • Coin, dice और card जैसे simple experiments के लिए बहुत उपयोगी।

Practice

(1/5)
1. A fair coin is tossed once. What is the probability of getting a Head?
easy
A. 1/2
B. 1/3
C. 1
D. 0

Solution

  1. Step 1: Identify total outcomes

    For a single coin toss the sample space is {H, T} → total outcomes = 2.

  2. Step 2: Identify favourable outcomes

    Favourable outcome for Head = {H} → count = 1.

  3. Step 3: Apply formula

    P(Head) = Favourable / Total = 1/2.

  4. Final Answer:

    1/2 → Option A.

  5. Quick Check:

    P(Head) + P(Tail) = 1/2 + 1/2 = 1 ✅
Hint: Count outcomes in the sample space first, then count favourable ones.
Common Mistakes: Forgetting that a fair coin has exactly 2 equally likely outcomes.
2. A fair six-sided die is rolled once. What is the probability of getting an even number?
easy
A. 1/2
B. 1/3
C. 1/6
D. 2/3

Solution

  1. Step 1: Identify total outcomes

    A die has faces {1,2,3,4,5,6} → total outcomes = 6.

  2. Step 2: Identify favourable outcomes

    Even numbers = {2,4,6} → favourable count = 3.

  3. Step 3: Apply formula

    P(even) = 3 / 6 = 1/2.

  4. Final Answer:

    1/2 → Option A.

  5. Quick Check:

    There are 3 even and 3 odd faces, so probability of even = 3/6 = 1/2 ✅
Hint: List outcomes quickly: count evens vs total faces.
Common Mistakes: Counting only 2 even numbers (forgetting 6) or dividing incorrectly.
3. One card is drawn at random from a standard 52-card deck. What is the probability that the card is a Heart?
easy
A. 1/2
B. 1/13
C. 1/4
D. 3/4

Solution

  1. Step 1: Identify total outcomes

    A standard deck has 52 cards → total outcomes = 52.

  2. Step 2: Identify favourable outcomes

    There are 13 hearts in the deck → favourable count = 13.

  3. Step 3: Apply formula

    P(Heart) = 13 / 52 = 1/4.

  4. Final Answer:

    1/4 → Option C.

  5. Quick Check:

    13 hearts out of 52 → dividing by 13 gives 1/4 ✅
Hint: Remember each suit has 13 cards → probability for any suit = 13/52 = 1/4.
Common Mistakes: Using 26 (red cards) instead of 13 (hearts) when asked for a specific suit.
4. A bag contains 3 red balls and 2 blue balls. One ball is drawn at random. What is the probability that the drawn ball is red?
medium
A. 2/5
B. 3/5
C. 1/2
D. 3/4

Solution

  1. Step 1: Identify total outcomes

    Total balls = 3 red + 2 blue = 5.

  2. Step 2: Identify favourable outcomes

    Favourable (red) = 3.

  3. Step 3: Apply formula

    P(red) = 3 / 5 = 3/5.

  4. Final Answer:

    3/5 → Option B.

  5. Quick Check:

    3 red out of 5 total → 3/5 = 0.6 ✅
Hint: Add counts for total, then place the favourable count over total.
Common Mistakes: Swapping counts (using blue count instead of red) or forgetting to sum total balls first.
5. Two fair coins are tossed. What is the probability of getting at least one Head?
medium
A. 1/4
B. 1/2
C. 3/8
D. 3/4

Solution

  1. Step 1: Identify total outcomes

    For two coins the sample space is {HH, HT, TH, TT} → total outcomes = 4.

  2. Step 2: Identify favourable outcomes

    At least one Head = {HH, HT, TH} → favourable count = 3.

  3. Step 3: Apply formula

    P(at least one Head) = 3 / 4 = 3/4.

  4. Final Answer:

    3/4 → Option D.

  5. Quick Check:

    Only TT has no head → 1 outcome without head → 1/4 no-head, so at least one head = 1 - 1/4 = 3/4 ✅
Hint: Sometimes easier to find complement: P(at least one Head) = 1 - P(no Heads).
Common Mistakes: Counting HT and TH as the same outcome (they are distinct), leading to wrong totals.

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