Chances of getting a specific combination of playing cards.

Question

Suppose I have 3 playing cards: 2,3,4

If I were to take 2 cards from an infinte deck I will get these unique combinations:

• 2,2
• 2,3
• 2,4
• 3,3
• 3,4
• 4,4

Now to the question. If I wanted to know what the chances are of getting one particular combination (say 2,3) do I need to include the double occurrence (3,2 which I omitted above)? Which would result in a 2/9 chance rather than a 1/9 chance.

Answer 1

If you wanted to determine the chances of getting one particular combination (say 2,3) you DO need to include the double occurrence (3,2).

Let's just look at the probability tree for just $2$ and $3$:

$$All~Cards~~~~~~~~$$$$/|~~~~~~~~$$$$2~~~~~3~~~(1/3)$$$$/|~~~~/|~~~~~~~~~~~~~~$$$$2~~3~~~~2~~3~~(1/3)~~~~~~~$$ As you can see, at each step, we still have a chance to get the conjugate group of cards. So, if we want to see our chances of getting both a $2$ and $3$ from the infinite deck, we would do $\frac13\cdot\frac13+\frac13\cdot\frac13=\frac29$.

Thus, the probability is $2/9$.