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Coin and Dice Based Probability

Introduction

Coin and dice experiments are the simplest and most frequent probability models. They use equally likely outcomes and help build intuition for sample spaces, counting outcomes, and basic probability formulas. Mastering these patterns prepares you for multi-stage experiments and conditional probability.

Typical questions ask for probabilities of single outcomes (e.g., head on a coin), combined outcomes (e.g., sum on two dice), or events like "at least one", "exactly k", and complements.

Pattern: Coin and Dice Based Probability

Pattern

Use the equally likely outcomes model: list the sample space, count favourable outcomes, then apply P(E) = Favourable / Total.

Key reminders:

  • Single fair coin → sample space = {H, T} (2 outcomes).
  • n fair coins → total outcomes = 2ⁿ (ordered outcomes count separately).
  • 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 are tossed. Find the probability of getting exactly one Head.
(ii) Two fair dice are rolled. Find the probability that the sum is 7.

Solution

  1. Step 1: List all outcomes for two coins

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

  2. Step 2: Identify favourable outcomes for exactly one Head

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

  3. Step 3: Compute probability of exactly one Head

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

  4. Step 4: Verify using combinations

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

  5. Step 5: List total outcomes for two dice

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

  6. Step 6: Identify pairs giving a sum of 7

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

  7. Step 7: Compute probability of sum = 7

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

  8. Final Answer:

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

  9. Quick Check:

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

Quick Variations

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

2. “Exactly k” Heads → Use nCk × (1/2)ⁿ for coins.

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

Trick to Always Use

  • Step 1: Identify if outcomes are ordered (coins or dice).
  • Step 2: Use total = 2ⁿ (coins) or 6ᵐ (dice).
  • Step 3: Apply complement for “at least” or “none” type questions.

Summary

Summary

  • Formula: P(E) = Favourable outcomes ÷ Total outcomes.
  • For n coins → total outcomes = 2ⁿ.
  • For m dice → total outcomes = 6ᵐ.
  • Use combinations or direct counting for multiple outcomes.
  • Apply complement rule for “at least one” type problems.
  • Always verify that probabilities sum to 1 for consistency.

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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