Question

Three unbiased coins are tossed simultaneously. Find the probability of getting:

1. at least one head 
2. exactly one tail 
3. two heads and one tail 
4. at most two heads

Answer 1

Total Possible Outcomes: When three unbiased coins are tossed simultaneously, the sample space is:

\[ S = {\text{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}} \]

\[ n(S) = 8 \]

(i) At least one head: All outcomes except TTT have at least one head.

Favourable outcomes: `({\text{HHH, HHT, HTH, THH, HTT, THT, TTH}})`

(7 outcomes) 

\[ P(\text{at least one head}) = \dfrac{7}{8} \]

(ii) Exactly one tail: Favourable outcomes: `({\text{HHT, HTH, THH}} )`

(3 outcomes)

\[ P(\text{exactly one tail}) = \dfrac{3}{8} \]

(iii) Two heads and one tail: Favourable outcomes: `({\text{HHT, HTH, THH}})`

(3 outcomes)

\[ P(\text{two heads and one tail}) = \dfrac{3}{8} \]

(iv) At most two heads: Possible outcomes can have 0, 1 or 2 heads (all outcomes except HHH).

Favourable outcomes: `({\text{HHT, HTH, THH, HTT, THT, TTH, TTT}})` 

(7 outcomes) 

\[ P(\text{at most two heads}) = \dfrac{7}{8} \]