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Coin and Dice Based Probability

Introduction

coin மற்றும் dice experiments என்பது மிகவும் எளிய மற்றும் அடிக்கடி வரும் probability models ஆகும். இவை equally likely outcomes-ஐ பயன்படுத்தி, sample space, outcomes எண்ணல், மற்றும் அடிப்படை probability formulas-க்கு நல்ல புரிதலை உருவாக்க உதவுகின்றன. இவற்றை நன்றாக கற்றுக்கொண்டால், multi-stage experiments மற்றும் conditional probability போன்ற தலைப்புகளுக்கு தயார் ஆகலாம்.

பொதுவான கேள்விகள் single outcomes (எ.கா., coin-இல் head), combined outcomes (எ.கா., two dice-இன் sum), அல்லது “at least one”, “exactly k”, மற்றும் complements போன்ற events-ஐப் பற்றி இருக்கும்.

Pattern: Coin and Dice Based Probability

Pattern

equally likely outcomes model-ஐ பயன்படுத்தவும்: sample space-ஐ பட்டியலிட்டு, favourable outcomes-ஐ எண்ணி, பின்னர் P(E) = Favourable / Total பயன்படுத்தவும்.

முக்கிய நினைவூட்டல்கள்:

  • Single fair coin → sample space = {H, T} (2 outcomes).
  • n fair coins → total outcomes = 2ⁿ (ordered outcomes தனித்தனியாக எண்ணப்படும்).
  • 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 toss செய்யப்படுகின்றன. exactly one Head கிடைக்கும் probability-ஐ கண்டறியவும்.
(ii) Two fair dice roll செய்யப்படுகின்றன. sum = 7 ஆகும் probability-ஐ கண்டறியவும்.

Solution

  1. Step 1: two coins-க்கான எல்லா outcomes-ஐ பட்டியலிடவும்

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

  2. Step 2: exactly one Head-க்கு favourable outcomes-ஐ கண்டறியவும்

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

  3. Step 3: exactly one Head-ன் probability-ஐ கணக்கிடவும்

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

  4. Step 4: combinations மூலம் verify செய்யவும்

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

  5. Step 5: two dice-க்கான total outcomes-ஐ கண்டறியவும்

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

  6. Step 6: sum = 7 தரும் pairs-ஐ அடையாளம் காண்க

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

  7. Step 7: sum = 7-ன் probability-ஐ கணக்கிடவும்

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

  8. Final Answer:

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

  9. Quick Check:

    (i) 4-இல் 2 favourable → 1/2
    (ii) 36-இல் 6 favourable → 1/6 ✅

Quick Variations

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

2. “Exactly k” Heads → coins-க்கு nCk × (1/2)ⁿ பயன்படுத்தவும்.

3. Dice patterns → even sum, doubles, total divisibility போன்றவை.

Trick to Always Use

  • Step 1: outcomes ordered ஆவையா (coins அல்லது dice) என்பதை கண்டறியவும்.
  • Step 2: total outcomes = coins-க்கு 2ⁿ, dice-க்கு 6ᵐ பயன்படுத்தவும்.
  • Step 3: “at least” அல்லது “none” வகை கேள்விகளுக்கு complement rule பயன்படுத்தவும்.

Summary

Summary

  • Formula: P(E) = Favourable outcomes ÷ Total outcomes.
  • n coins → total outcomes = 2ⁿ.
  • m dice → total outcomes = 6ᵐ.
  • multiple outcomes-க்கு combinations அல்லது direct counting பயன்படுத்தவும்.
  • “at least one” வகை பிரச்சினைகளுக்கு complement rule பயன்படுத்தவும்.
  • consistency-க்காக probabilities-ன் கூட்டுத்தொகை 1 ஆகுமா என்பதை எப்போதும் சரிபார்க்கவும்.

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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