There are 4 cards in a game, numbered 1 to 4. A player randomly draws one card at a time (and without replacement). The game ends when the numbers on the drawn cards sum to 3 and above.

Let X be a random variable - The sum of numbers on the drawn cards.

Let Y be a RV - The number of cards the player had drawn.

Find the joint probability of X & Y.

Well, my struggle is mostly in which method do I calculate the probabilities. I know that X can take the values 3,4,5,6 and Y can take 1,2.

The sample space is 8 I think, because there are 8 possible scenarios of taking cards out until we stop. Am I right to think this way?

However when putting the probabilities in the table, they sum up to 6/8 only. What am I overlooking?

Thanks a bunch!