You draw two cards, simultaneously, from a 52-card deck. What is the probability that at least one is an ace?

bigslick

For one card, this is easy: the odds are one in thirteen, for there are four aces in 52 cards, and 4/52 = 1/13.

With a second card drawn at the same time, we must consider the 12/13ths of the time that the first card drawn is not an ace. When this happens, 51 cards remain, with four of them aces, so there is an additional  4/51sts of this 12/13ths that must be added to the 1/13th for the first card drawn.

Therefore, the odds of drawing at least one ace, in two cards drawn from a standard deck, are 1/13 + (4/51)(12/13) = (1/13)(51/51) + (4/51)(12/13) = (51 + 48)/[(51)(13)] = 99/663 = 33/221, or 33 out of 221 attempts, which is as far as the fraction will reduce. In decimal form, as a percentage, this happens ~14.93% of the time.

If I made an error above, please let me know in a comment. I do not claim to be infallible.

[Image credit: I found the image above here.]

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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One Response to You draw two cards, simultaneously, from a 52-card deck. What is the probability that at least one is an ace?

  1. teachezwell says:

    I wouldn’t imagine challenging your math! I also appreciate the way you laid out your reasoning.

    Like

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