June Puzzler - Digital Agent Blog
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# June Puzzler

## 27 Jun June Puzzler ### Digital Agent engineers solve complex puzzles for our customers every day. Can you solve the Puzzle we created for you?

QUESTION:

A standard deck with 52 cards—26 red and 26 black—is shuffled.
What is the probability that the top card is the same color as the bottom card?

A. more than 50%
B. less than 50%
C. exactly 50%

There are a few ways to rationalize this problem. First by logic… If the top card is red, there are 25 other red cards and 26 black cards that the bottom card could possibly be, so black is slightly more likely.

If the top card is black, we use the same logic to determine that a red card on the bottom is more likely, so there is less than 50% chance the top and bottom cards are the same color.

Check out how one submitter calculated the probability mathematically…

The likelihood of the first 2 pulls being the same color were always as follows (assumes an even number of overall items):

c = number of items of one color
t = the total number of items of both colors (which can also just be 2c)
P = probability of the first 2 items pulled being the same color
P = (The sum of 1+2+3…(c-1)) / (The sum of 1+2+3…(t-1)    P =  24.5% likelihood that both the top and bottom cards are black.  Because we want to know the probability for the red+red AND black+black combinations, double P to get the exact answer: less than 50%

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