[Math] 2 Cards are drawn from a deck of cards. What is the probability of having drawn a black card OR an ace

probability

I'm sure this question involves combinations, dividing the probabilities of each black card/ace over ${52}\choose{2}$ (for $2$ cards drawn) and adding them but I'm not getting the right answer. The given solution is: $\frac{183}{221}$.

Thanks!

Best Answer

You are looking for the probability of drawing two cards that include a black card or an ace. When you are looking for the probability where multiple events can trigger the desired outcome (black card OR ace, AT LEAST one of the dice is a six, etc) sometimes it is easier to find the probability of the opposite happening and subtracting that from 100%.

In this case, the opposite is drawing two cards that are red and not an ace. There are 24 cards that fit this criteria, so the only way to not get a black card or ace is to get two cards, both of which were from those 24 cards.

$$1-\frac{24\choose2}{52\choose2}=\frac{175}{221}$$

Another way to think of it would be

$$\frac{{52\choose2}-{24\choose2}}{52\choose2}$$

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