Coin Toss Probability: Single, Double, and Triple Coins for CBSE Class 10

From the opening kickoff of a FIFA World Cup final to the coin toss of an international cricket match, a flipped coin is humanity's universal symbol of perfect fairness. Why? Because a standard coin has two symmetrical, mutually exclusive faces—Heads ($H$) and Tails ($T$)—each having an identical $50\%$ chance of landing face up.

In CBSE Class 10 Mathematics, Chapter 14 (Probability), coin toss experiments form the classic introduction to multi-stage sample spaces. Mastering the tree of outcomes for one, two, and three coins, and avoiding the notorious linguistic trap between "at least" and "at most", is essential for scoring full marks.

What You Will Learn

• Sample space and probabilities for tossing a single coin ($n = 2$)
• Sample space for tossing two coins simultaneously ($n = 4$)
• The crucial linguistic difference: "At least" vs. "At most"
• Sample space for tossing three coins simultaneously ($n = 8$)
• The systematic binary listing technique ($4H-4T, 2H-2T, 1H-1T$)
• Solving the classic NCERT Hanif Game Problem
• Board exam tips, presentation formats, and common student errors

1. Tossing a Single Coin ($n = 2$)

When an unbiased coin is tossed once:

• Sample Space: $S = \{H, T\}$
• Total number of possible outcomes: $n(S) = \mathbf{2}$.
• Probability of getting a Head: $P(H) = \frac{1}{2} = 0.5$
• Probability of getting a Tail: $P(T) = \frac{1}{2} = 0.5$
• Both outcomes are equally likely elementary events ($P(H) + P(T) = 1$).

2. Tossing Two Coins Simultaneously ($n = 4$)

Whether you toss two identical coins simultaneously or toss one single coin twice in succession, the set of possible outcomes is mathematically identical.

The Sample Space:

$S = \{HH, \; HT, \; TH, \; TT\}$ $\mathbf{n(S) = 2^2 = 4}$

                               Coin 1        Coin 2        Outcome
                                 H ----------> H  ------->  HH  (2 Heads)
                                 |
                                 +-----------> T  ------->  HT  (1 Head, 1 Tail)
                                 
                                 T ----------> H  ------->  TH  (1 Tail, 1 Head)
                                 |
                                 +-----------> T  ------->  TT  (0 Heads)

Important: <u>$HT$ and $TH$ are two distinct outcomes! $HT$ means Coin 1 was Heads and Coin 2 was Tails; $TH$ means Coin 1 was Tails and Coin 2 was Heads. Never treat them as the same outcome!</u>

3. The Great Board Exam Trap: "At Least" vs. "At Most"

In board exams, students frequently mix up these two English phrases:

    Phrase               Mathematical Meaning                     For Two Coins (Max = 2 Heads)
    ---------------------------------------------------------------------------------------------
    "At least ONE head"   Greater than or equal to 1 (≥ 1)         1 or 2 Heads  → {HT, TH, HH} = 3
    "At most ONE head"    Less than or equal to 1 (≤ 1)            0 or 1 Head   → {TT, HT, TH} = 3
    "Exactly ONE head"    Strictly equal to 1 (= 1)                1 Head only   → {HT, TH} = 2

1. Probability of getting at least one head: Favourable outcomes $= \{HT, TH, HH\} \implies n(E) = 3$. $\mathbf{P(\text{At least 1 Head}) = \frac{3}{4}}$
2. Probability of getting at most one head: Favourable outcomes $= \{TT, HT, TH\} \implies n(E) = 3$. $\mathbf{P(\text{At most 1 Head}) = \frac{3}{4}}$
3. Probability of getting exactly two heads: Favourable outcome $= \{HH\} \implies n(E) = 1$. $\mathbf{P(2\text{ Heads}) = \frac{1}{4}}$

4. Tossing Three Coins Simultaneously ($n = 8$)

When three fair coins are tossed simultaneously: $\mathbf{n(S) = 2^3 = 8}$

The Systematic Binary Listing Technique:

To write all 8 outcomes without missing a single combination, use the binary halving pattern:

• First position: Write 4 Heads, then 4 Tails ($H, H, H, H, T, T, T, T$).
• Second position: Alternate in pairs ($HH, TT, HH, TT$).
• Third position: Alternate singly ($H, T, H, T, H, T, H, T$).

The Complete Sample Space:

$\mathbf{S = \{HHH, \; HHT, \; HTH, \; THH, \; HTT, \; THT, \; TTH, \; TTT\}}$

5. Solved CBSE Board Examination Problems

Solved Example 1: Three Coins Analysis

Problem: Three fair coins are tossed together. Find the probability of getting: (i) exactly two heads
(ii) at least two heads
(iii) at most two heads
(iv) no heads

Solution: Total possible outcomes $n(S) = 8$.

1. (i) Exactly two heads: Favourable outcomes have exactly $2$ $H$'s: $\{HHT, HTH, THH\} \implies n(E_1) = 3$. $P(\text{Exactly 2 Heads}) = \mathbf{\frac{3}{8}}$

2. (ii) At least two heads: Means $2$ or $3$ heads: $\{HHT, HTH, THH, HHH\} \implies n(E_2) = 4$. $P(\text{At least 2 Heads}) = \frac{4}{8} = \mathbf{\frac{1}{2}}$

3. (iii) At most two heads: Means $0, 1,$ or $2$ heads (everything EXCEPT $HHH$!): Favourable outcomes $= 8 - 1 = 7$. $P(\text{At most 2 Heads}) = \mathbf{\frac{7}{8}}$

4. (iv) No heads: Favourable outcome is all tails: $\{TTT\} \implies n(E_4) = 1$. $P(\text{No Heads}) = \mathbf{\frac{1}{8}}$

Solved Example 2: The Hanif Game Problem (NCERT Classic)

Problem: A game consists of tossing a one-rupee coin $3$ times and noting its outcome each time. Hanif wins if all the tosses give the same result, i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.

Solution:

1. Total possible outcomes when a coin is tossed 3 times: $S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\} \implies n(S) = 8$
2. Hanif wins when the result is three heads or three tails: $\text{Winning outcomes} = \{HHH, TTT\} \implies n(\text{Win}) = 2$ $P(\text{Win}) = \frac{2}{8} = \frac{1}{4}$
3. Hanif loses if he gets any other combination: $\text{Losing outcomes} = \{HHT, HTH, THH, HTT, THT, TTH\} \implies n(\text{Lose}) = 6$ $P(\text{Lose}) = \frac{6}{8} = \mathbf{\frac{3}{4}}$ (Alternatively, using complementary probability: $P(\text{Lose}) = 1 - P(\text{Win}) = 1 - \frac{1}{4} = \frac{3}{4}$).
4. Therefore, <u>the probability that Hanif will lose the game is $\frac{3}{4}$ (or $0.75$)</u>.

6. Summary and Examination Tips

Exam Tip: In board exams, always write down the complete list of sample space elements at the start of your answer! Showing the sample space explicitly guarantees full method marks even if a minor arithmetic slip occurs later.

Common Mistake: Thinking "at most two heads" means strictly two heads. "At most" means the upper ceiling is 2 (so 0 heads, 1 head, and 2 heads are all valid favourable outcomes)!