What is the probability of getting a black card in a deck of 52 cards?

If given a standard deck of 52 cards, what is the probability of drawing a black card on the 1st, 2nd, 3rd, 4th ... $n^{th}$ draw?

I understand that the first is $26/52$, but the second gets a bit more complicated because there is two scenarios:
a) The first draw is black so the probability is $26/52$ * $25/51$
b) The first draw is non-black, so the probability is $26/52$ * $26/51$

I'm not sure what the formulaic expression of this is and how to consider both of these options in one calculation. I know it uses combinatorics, but I'm lost.

Thanks!

a card is randomly selected from a deck of card. Find the probability that it is a black card or a face card

5 years ago

There are 52 cards in a deck in total. Of those 52 cards, there are four different suits (diamonds, hearts, clubs, spades). There are 13 cards in each of the different suits. Also, there are 3 face cards in each of the different suits (therefore, there are 12 face cards in total). Diamonds and Hearts are red cards (there are 26 total red cards) and Clubs and Spades are black cards (there are 26 total black cards).

There are 26 black cards and 12 face cards in total. However, of those 26 black cards, there are 6 face cards. That means there are 26+12-6 = 32 cards in total that are either a black card or a face card, but not both. That means the answer to the question is 32/52 = 8/13.

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5 years ago

A = number of black cards B = number of picture cards (or face cards) C = number of black picture cards

There are 26 black cards (spades and clubs), so A = 26.

There are 3 picture cards (Jack, Queen, King) in each suit. There are 4 suits (clubs, hearts, spades, diamonds). So there are 3*4 = 12 picture cards. This means B = 12

There are 2 suits which are black (spades and clubs) with 3 face cards per suit, so 2*3 = 6 cards which are both black cards and face cards. So C = 6

The number of cards that are either black cards or a face card, or both, is...

D = A+B-C D = 26+12-6

D = 32

So there are 32 cards that are either black cards or a face card, or both

This is out of 52 cards total, so

probability of selecting a black card or a picture card = D/52

probability of selecting a black card or a picture card = 32/52

probability of selecting a black card or a picture card = 8/13

The final answer, as a fraction, is 8/13

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