Raised Fist0

Coin and Dice Based Probability

Start learning this pattern below

Jump into concepts and practice - no test required

or
Recommended
Test this pattern10 questions across easy, medium, and hard to know if this pattern is strong

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: Coin and Dice Based Probability

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

  • 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.