Three coins are tossed together. Write all the possible outcomes. Now, find the probability of getting :

(i) exactly two heads.

(ii) at least two heads.

(iii) atmost two heads.

(iv) all tails

(v) at least one tail.

When three coins are tossed simultaneously;

Possible outcomes are : {HHH, TTT, HHT, HTH, THH, TTH, THT, HTT}.

(i) Favourable outcomes for getting exactly two heads are : HHT, HTH, THH.

No. of favourable outcomes = 3

P(getting exactly two heads) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{3}{8}$.

Hence, the probability of getting exactly two heads = $\dfrac{3}{8}$.

(ii) Favourable outcomes for getting at least two heads are : HHT, HTH, THH, HHH.

No. of favourable outcomes = 4

P(getting atleast two heads) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{4}{8} = \dfrac{1}{2}$.

Hence, the probability of getting at least two heads = $\dfrac{1}{2}$.

(iii) Favourable outcomes for getting at most two heads are : TTT, HHT, HTH, THH, TTH, THT, HTT.

No. of favourable outcomes = 7

P(getting atmost two heads) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{7}{8}$.

Hence, the probability of getting at most two heads = $\dfrac{7}{8}$.

(iv) Favourable outcomes for getting all tails is : TTT.

No. of favourable outcomes = 1

P(getting all tails) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{1}{8}$.

Hence, the probability of getting all tails = $\dfrac{1}{8}$.

(v) Favourable outcomes for getting at least one tail is : TTT, HHT, HTH, THH, TTH, THT, HTT.

No. of favourable outcomes = 7

P(getting at least one tail) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{7}{8}$.

Hence, the probability of getting at least one tail is = $\dfrac{7}{8}$.