Question

Two coins are tossed simultaneously. The probability of getting at least one tail is \dots

• A. \( \frac{1}{4} \)
• B. \( \frac{3}{4} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{3}{8} \)

Hint:

For probability questions containing the words “at least one”, try using the complement method: \[ P(\text{at least one}) = 1 - P(\text{none}) \] It is usually faster and reduces mistakes.

Answer 1

The Correct Option is B

Concept: Probability measures the chance of occurrence of an event. The probability of an event \(E\) is given by: \[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \] When coins are tossed, each coin has two possible outcomes: \[ H \text{ (Head)} \quad \text{or} \quad T \text{ (Tail)} \] For two coins tossed simultaneously, we construct the complete sample space.

Step 1: Write all possible outcomes. When two coins are tossed, possible outcomes are: \[ S = \{HH,\ HT,\ TH,\ TT\} \] where:
• \(HH\): both heads
• \(HT\): first head, second tail
• \(TH\): first tail, second head
• \(TT\): both tails

Step 2: Count total outcomes. Total number of outcomes: \[ n(S) = 4 \]

Step 3: Understand the phrase “at least one tail”. “At least one tail” means:
• exactly one tail, or
• two tails So favorable outcomes are: \[ E = \{HT,\ TH,\ TT\} \]

Step 4: Count favorable outcomes. \[ n(E) = 3 \]

Step 5: Apply probability formula. \[ P(E) = \frac{n(E)}{n(S)} \] Substitute values: \[ P(E) = \frac{3}{4} \]

Step 6: Alternative verification method. Sometimes “at least one” problems become easier using complement rule. The opposite of “at least one tail” is: \[ \text{No tails} \] The only outcome with no tail is: \[ HH \] So: \[ P(HH) = \frac{1}{4} \] Therefore: \[ P(\text{at least one tail}) = 1 - \frac{1}{4} = \frac{3}{4} \] This confirms the answer. Final Answer: \[ \boxed{\frac{3}{4}} \]