2 Cards are drawn from a deck of cards.

Question

I use this site for learning mathematics. I have come across this question. and the explanations are very much clear. Suppose the question is

2 Cards are drawn from a deck of cards. What is the probability that one of them is a black card and one of them is ace?

Can anybody help me to get the intuition for this problem,

Answer 1

Hint: Try working out the total number of ways that you could be given a black card and an ace in the 2 cards that you take from the pack. For example if we assume that one of the aces is the ace of hearts, then there are 26 black cards which could accompany it, meaning there are 26 combinations when we assume we get the ace of hearts. We can continue, assuming we have a different ace, then adding all of these together and dividing through by the total number of 2 card hands we could have, we will get the probability.

Edit, added extras: To calculate probability we want to calculate how many ways we can obtain what we want (in this case one of the cards to be black and the other to be an ace) and divide through by the total number of possibilities (in this case $ {52 \choose 2} $ since this is the total number of 2 card hands we could be given).

So we want to count how many ways we could have an ace and a black card. Now, we can partition by which ace we have. If we have the ace of hearts then as explained above there are 26 black cards to accompany it giving 26 possibilities. Likewise if we have the ace of diamonds there are also 26. However if we had the ace of clubs then we have used one of the black cards in our assumption on the ace, which means there are now only 25 black cards left to be the other card. Can you see where to go from here?

Answer 2

Here is a picture:

The first card appears left-right, the second card top-bottom in lighter colors.

The colored squares represent picking either an ace or black card, or both. The yellow squares are no-goes.

I count $2100$ colored (not yellow) squares, and there are $2652$ available squares in total (the grid is $52\times52$), so the probability is $\dfrac{175}{221}$.