Yes, you can win it. Just think about what you have already done with the girls prior to the game show when choosing who competes. Partial spoiler explanation below.
You raced with Lauren in the park, so choose her for the race competition (Tortoise and Hare). You got a quiz from Judie in the library, so choose her for quiz (Rain of Knowledge). Carla is into such things, so choose her for Mentalist.