Two events are mutually exclusive when the intersection of them is `phi`.
Two events are mutually exhaustive when the union of them is sample space.
In the given question, Sample space, `S = {(H,H,H),(H,H,T),(H,T,T),(T,H,H),(H,T,H),(T,H,T),(T,T,H),(T,T,T)}`
Here, `H` represents Head and `T` represents Tail.
So,
(i)Two events which are mutually exclusive.
`A =` All three Heads, `B` = All three Tails
As, `AnnB= phi`
So, `A` and `B` are mutually exclusive events.
(ii)Three events which are mutually exclusive and exhaustive.
`A = {(H,H,H),(T,T,T)}`
`B= {(H,H,T),(H,T,H),(T,H,H)}`
`C= {(T,T,H),(T,H,T),(H,T,T)}`
As, `AnnBnnC = phi`
`AuuBuuC = S`
Events `A,B,C` are mutually exclusive and exhaustive.
(iii)Two events, which are not mutually exclusive.
`A = {H,H,T}`
`B= {H,T,H}`
As both these events have two heads and one tail, these events are not mutually exclusive.
(iv)Two events which are mutually exclusive but not exhaustive.
`A =` All three Heads, `B` = All three Tails
As, `AnnB= phi`
`AuuB != S`
So, `A` and `B` are mutually exclusive but not mutually exhaustive events.
(v)Three events which are mutually exclusive but not exhaustive.
`A = {(H,H,H)}`
`B= {(H,H,T)}`
`C= {(H,T,T)}`
Here, `AnnBnnC = phi`
`AuuBuuC !=S`
So, all these three events are mutually exclusive bot not mutually exhaustive.
