Question

Two different coins are tossed simultaneously. What is the probability of getting :
(i) at least one head ?
(ii) at most one tail ?
(iii) a head and a tail ?

Hint:

"At least one head" and "At most one tail" in a two-coin toss describe the exact same set of outcomes. Writing down the sample space clearly is the safest way to avoid confusion with words like "at least" and "at most".

Answer 1

We are asked to find probabilities when two different coins are tossed simultaneously.
Step 1: Sample Space
- Let the two coins be Coin 1 and Coin 2.
- Each coin can land as Head (H) or Tail (T).
- The sample space for two coins is:
\[ S = \{ (H, H), (H, T), (T, H), (T, T) \} \]
- Total number of outcomes, \(n(S) = 4\).
(i) Probability of at least one head
- "At least one head" means 1 or 2 heads appear.
- Favorable outcomes: \((H, H), (H, T), (T, H)\)
- Number of favorable outcomes = 3
\[ P(\text{at least one head}) = \frac{3}{4} \]
(ii) Probability of at most one tail
- "At most one tail" means 0 or 1 tail appears.
- Outcomes with 0 tail: \((H, H)\)
- Outcomes with 1 tail: \((H, T), (T, H)\)
- Total favorable outcomes = 3
\[ P(\text{at most one tail}) = \frac{3}{4} \]
(iii) Probability of a head and a tail
- "A head and a tail" means one coin shows H and the other shows T.
- Favorable outcomes: \((H, T), (T, H)\)
- Number of favorable outcomes = 2
\[ P(\text{head and tail}) = \frac{2}{4} = \frac{1}{2} \]
Answer:
(i) \(P = \frac{3}{4}\)
(ii) \(P = \frac{3}{4}\)
(iii) \(P = \frac{1}{2}\)