The answer given is $\displaystyle 0.5*0.5[(0.4*0.3*0.6)+(0.6*0.6)+(0.6*0.4*0.3)] $

however i think that we do not have to $\displaystyle *0.5*0.5$

it is already given that sam won the first set and joseph won the second set.

if we were to take into account the outcome of the first 2 sets, then we'll have to consider cases such as "joseph winning the first set then sam winning the second set", and "joseph winning both the first and second set" and "sam winning both the first and second set" too. these all can lead to joseph winning the game.

if we do not take into account the other possible outcomes of first 2 set but still $\displaystyle *0.5*0.5$, this probability tree does not give a total probability of 1. but only 0.25

that's why i think we should not $\displaystyle \times0.5\times0.5$

your opinions?