examday

/TMUA

Nine cards are numbered $1,2,\ldots,9$. Three are drawn without replacement, with every selection of three cards equally likely.

After seeing these three cards, a player must discard either one or two of them. The discarded cards are not returned to the pack. The same number of cards is then drawn from the six undrawn cards, without replacement, with every selection of that size equally likely.

The player wins if the sum of the final three numbers is divisible by $3$. What is the greatest possible probability of winning, when the player chooses which cards to discard as favourably as possible?