Hi there.

probability(no one recovers and the game is won) = 2/3 * 2/3 *1/16

prob(one guy recovers and the game is won) = 2/3*1/3 * 1/2 + 1/3 * 2/3 *1/2

prob(everyone recovers and the game is won) = 2/3*2/3 * 3/4

prob("game is won")=2/3 * 2/3 *1/16 + 2/3*1/3 * 1/2 + 1/3 * 2/3 *1/2 + 2/3*2/3 * 3/4

i just added all three probabilities and i got prob() = 0.5833....3