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Coin and Dice Based Probability

Introduction

Coin और dice वाले experiments सबसे simple और सबसे common probability models होते हैं। इनमें equally likely outcomes होते हैं, और ये sample spaces, outcomes को count करना, और basic probability formulas की समझ को strong बनाते हैं। इन patterns को सीखने से आप multi-stage experiments और conditional probability के लिए भी तैयार हो जाते हैं।

Typical questions में single outcomes (जैसे coin पर head), combined outcomes (जैसे दो dice का sum), या “at least one”, “exactly k”, और complement events पूछे जाते हैं।

Pattern: Coin and Dice Based Probability

Pattern

Equally likely outcomes model का उपयोग करें: पहले sample space लिखें, favourable outcomes गिनें, फिर formula P(E) = Favourable / Total लगाएँ।

Key reminders:

  • Single fair coin → sample space = {H, T} (2 outcomes).
  • n fair coins → total outcomes = 2ⁿ (ordered outcomes अलग माने जाते हैं)।
  • One fair die → sample space = {1,2,3,4,5,6} (6 outcomes).
  • Two dice (ordered) → total outcomes = 6 × 6 = 36.

Step-by-Step Example

Question

(i) Two fair coins को toss किया जाता है। Exactly one Head आने की probability निकालें।
(ii) Two fair dice roll किए जाते हैं। Sum = 7 आने की probability निकालें।

Solution

  1. Step 1: Two coins के सभी outcomes लिखें

    2 coins का sample space (ordered) = {HH, HT, TH, TT} → total outcomes = 4.

  2. Step 2: Exactly one Head के favourable outcomes पहचानें

    Exactly one Head outcomes = {HT, TH} → favourable = 2.

  3. Step 3: Exactly one Head आने की probability निकालें

    P(exactly one Head) = Favourable / Total = 2 / 4 = 1/2.

  4. Step 4: Combinations से verify करें

    Quick check: Using combinations → 2C1 × (1/2)² = 2 × 1/4 = 1/2 ✅

  5. Step 5: Two dice के total outcomes लिखें

    Two dice → total ordered pairs = 6 × 6 = 36.

  6. Step 6: Sum = 7 देने वाले pairs पहचानें

    Pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 favourable.

  7. Step 7: Sum = 7 आने की probability निकालें

    P(sum = 7) = 6 / 36 = 1/6.

  8. Final Answer:

    (i) 1/2
    (ii) 1/6

  9. Quick Check:

    (i) 4 में से 2 favourable → 1/2
    (ii) 36 में से 6 favourable → 1/6 ✅

Quick Variations

1. “At least one” problems → P(at least one Head) = 1 - P(no Heads).

2. “Exactly k” Heads → nCk × (1/2)ⁿ का उपयोग करें।

3. Dice patterns → even sum, doubles, total divisibility आदि।

Trick to Always Use

  • Step 1: देखें कि outcomes ordered हैं या नहीं (coins और dice में ordered माने जाते हैं)।
  • Step 2: Coins के लिए total = 2ⁿ और dice के लिए total = 6ᵐ लगाएँ।
  • Step 3: “at least” या “none” type questions में complement rule का उपयोग करें।

Summary

Summary

  • Formula: P(E) = Favourable outcomes ÷ Total outcomes.
  • n coins के लिए total outcomes = 2ⁿ.
  • m dice के लिए total outcomes = 6ᵐ.
  • Multiple outcomes के लिए combinations या direct counting का उपयोग करें।
  • “At least one” type problems में complement rule helpful होता है।
  • हमेशा verify करें कि total probabilities का sum = 1 हो।

Practice

(1/5)
1. Two fair coins are tossed. What is the probability of getting two Tails?
easy
A. 1/4
B. 1/2
C. 3/4
D. 1

Solution

  1. Step 1: Identify total outcomes

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

  2. Step 2: Identify favourable outcomes

    Favourable outcome for two Tails = {TT} → favourable count = 1.

  3. Step 3: Apply formula

    P(two Tails) = Favourable / Total = 1 / 4 = 1/4.

  4. Final Answer:

    1/4 → Option A.

  5. Quick Check:

    There are 4 equally likely outcomes and only one has TT → 1/4 ✅
Hint: List ordered outcomes (HH, HT, TH, TT) - count TT only.
Common Mistakes: Treating HT and TH as the same outcome when listing possibilities.
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

    Die faces = {1,2,3,4,5,6} → total = 6.

  2. Step 2: Identify favourable outcomes

    Even faces = {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:

    Three even and three odd faces → probability = 3/6 = 1/2 ✅
Hint: Count evens directly: 2, 4, 6 → 3 out of 6 faces.
Common Mistakes: Forgetting one of the even faces (like 6) when counting.
3. Three fair coins are tossed. What is the probability of getting at least one Tail?
easy
A. 1/8
B. 7/8
C. 3/8
D. 1/2

Solution

  1. Step 1: Identify total outcomes

    Total ordered outcomes for 3 coins = 2³ = 8.

  2. Step 2: Find complement (no Tail)

    No Tail means all Heads (HHH) → P(all Heads) = 1 / 8.

  3. Step 3: Use complement rule

    P(at least one Tail) = 1 - P(all Heads) = 1 - 1/8 = 7/8.

  4. Final Answer:

    7/8 → Option B.

  5. Quick Check:

    Only HHH has no tail → 1/8 without tail, so 7/8 with at least one tail ✅
Hint: For 'at least one' questions, compute the 'none' case and subtract from 1.
Common Mistakes: Trying to count all favourable outcomes directly and missing combinations.
4. Two fair dice are rolled. What is the probability that both dice show odd numbers?
medium
A. 1/3
B. 1/6
C. 1/4
D. 1/2

Solution

  1. Step 1: Identify total outcomes

    Total ordered pairs = 6 × 6 = 36.

  2. Step 2: Identify favourable outcomes

    Odd faces on a die = {1,3,5} → 3 choices for each die, so favourable ordered pairs = 3 × 3 = 9.

  3. Step 3: Apply formula

    P(both odd) = 9 / 36 = 1/4.

  4. Final Answer:

    1/4 → Option C.

  5. Quick Check:

    Each die has probability 1/2 of odd → combined = 1/2 × 1/2 = 1/4 ✅
Hint: Count odd faces per die (3) and multiply: 3×3 = 9 favourable pairs.
Common Mistakes: Counting unordered pairs instead of ordered pairs and undercounting outcomes.
5. Two fair dice are rolled. What is the probability that the sum of the faces is 11?
medium
A. 1/9
B. 1/12
C. 1/6
D. 1/18

Solution

  1. Step 1: Identify total outcomes

    Total ordered pairs = 6 × 6 = 36.

  2. Step 2: Identify favourable outcomes

    Pairs summing to 11: (5,6) and (6,5) → favourable count = 2.

  3. Step 3: Apply formula

    P(sum = 11) = 2 / 36 = 1/18.

  4. Final Answer:

    1/18 → Option D.

  5. Quick Check:

    Only two ordered pairs give 11 → 2/36 simplifies to 1/18 ✅
Hint: List ordered pairs that add to the target sum to avoid missing cases.
Common Mistakes: Forgetting symmetric pairs like (6,5) when listing favourable outcomes.

Mock Test

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