Difficulty: Tricky

Miguel hopes to become a magnificent mathematical magician. For his first trick he wants to present a standard 52-card deck and dramatically draw a face card (Jack, Queen or King). If the deck is randomly shuffled, what is the probability that Miguel will draw a face card on the first try?

A standard 52 card deck has four suits (spades, clubs, diamonds, hearts). Each suit has the cards from 1 (or ace) to 10, and a Jack, a Queen and a King.

Bonus question: a magician never reveals their secrets, but Miguel is considering replacing some of the non-face cards in the deck with more face cards. How many cards would he need to replace to have a 52-card deck that gives him a 50% chance of drawing a face card on the first try?

Need a hint?

Ask yourself, how many total face cards are there in a standard 52-card deck?

Finally, remember that probability can be calculated as: (number of desired outcomes) divided by (total outcomes possible). With a 52-card deck, there are 52 outcomes possible.

Brainteaser answer

Miguel has a 3/13 or 0.23 or 23% chance of drawing a face card on his first try. And to boost that chance to 50%, he would need to take out 14 non-face cards and replace them with 14 face cards.

To find the probability, it is helpful to remember that probability can be calculated as: (number of desired outcomes) divided by (total outcomes possible).

Now we can tackle each part of that equation. Total outcomes possible is an easier to place to start. With a shuffled deck, any card could be the top card. Since there are 52 cards and each one could be Miguel’s first card, there are 52 possible outcomes.

Next, we need to calculate the number of desired outcome. Or in other words, we need to count the face cards!

There are four suits in a deck of cards, and each suit has three face cards – the Jack, Queen and King. If there are 3 face cards per suit and 4 suits in the deck, then 3 times 4 gives us 12 total face cards.

Now we can calculate the probability: (number of desired outcomes) divided by (total outcomes possible) equals 12 divided by 52 which equals 3/13 as a fraction, 0.23 as a decimal or 23% as a percentage.

Bonus question: Miguel wants a 52-card deck with a 50% chance of drawing a face card. So he’ll need half the deck to be face cards. And half of a 52 card deck is 26 cards

As we calculated above, a standard 52-card deck already comes with 12 face cards. To get to 26, Miguel needs to replace 14 non-face cards with 14 face cards. This way 26 face cards divided by 52 total cards = 26/52 = ½ or 0.5 or 50%.

Hopefully that does the trick!

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