[jay_shark]

Let Z be the event that no player can win twice .

P(AB|Z ) =P(ABZ)/P(Z)

P(Z)=1 since eventually we can continue this until no player can win twice .

P(ABZ) = n(ABZ)/s(ABZ) = 8/(4/10*8/10 + 6/10*7/10) =8/74

There are 8 ways that the event ABZ occur together .
Also 4 out of 10 times there will be no match 8 out of 10 times ; 6 out of 10 times , there will be no match 7 out of 10 times . This gives us 74 in the denominator .