# Two cards drawn - probability

Two cards are drawn successively from a pack of $52$ cards. Find the probability that the cards are picture cards (Jack,Queen and King), or spades, or both.

I did the normal $A \cup B$ formula, but then I realised that this will fall under conditional probability. I don't have any formula for that though.

• What's the probability that the first drawn card is of the form you want? Conditioned on a success with the first card, what is the probability that the second is also a card of the form you want? – lulu Sep 3 '18 at 13:09
• @lulu is it (22/52)*(21/51)? – user585380 Sep 3 '18 at 13:12
• You need to select two out of 22 cards. – prog_SAHIL Sep 3 '18 at 13:17
• Yes! and it is good to verify that this gives the same result as the (solid) method proposed by @prog_SAHIL – lulu Sep 3 '18 at 13:20

If neither are spades, they should both be faces and the number of cases is ${\binom{9}{2}}$. If one of them is spade, the other one should either be spade or non-spade face. The former case has ${\binom{13}{2}}$ and the latter has ${\binom{13}{1}\binom{9}{1}}$ number of possible cases which makes a total probability $$\dfrac{\binom{9}{2}+\binom{13}{2}+\binom{9}{1}\binom{13}{1}}{\binom{52}{2}}$$
• So you mean the case $1\heartsuit,K\spadesuit$ is valid huh? – Mostafa Ayaz Sep 3 '18 at 13:45