Let $A_i$ be the event of playing an opponent of type $i$.
$A_1$ = playing the half of the opponent pool where the probability of winning P (winning|$A_1$) = 0.3
$A_2$ = playing the half of the opponent pool type 2, where the probability of winning P (winning|$A_2$) = 0.4
$A_3$ = playing the half of the opponent pool type 3, where the probability of winning P (winning|$A_3$) = 0.5
By total probability theorem, we get, P(winning) = P($A_1$)P(winning|$A_1$) + P($A_2$)P(winning|$A_2$) + P($A_3$)P(winning|$A_3$)
P (winning) = 50% $\times$ 0.3 + 25% $\times$ 0.4 + 25% $\times$ 0.5
P (winning) = 0.375.