Solution to Puzzle #148: The First Black Ace

This was a bit of a  counterintuitive puzzle – four people sent the correct answers – Suman Saraf, Pratik Poddar, Abhinav Jain and Karan Sharma – well done!

The answer is the first card. It is a bit difficult to imagine intuitively that you should guess the first card in a pack of 52 cards as the one which has the highest probability of finding the first black ace. The challenge is that the probability is very low, but probability of it being in any other location is still smaller.

Here is how it works:

Probability that the first card is an ace is 2/52 (there are two possible black aces)

Probability that the second card is the first black ace is 50/52* 2/51 (please note that it is smaller than the first one by a very small factor – 50/51

…if we continue, the probability that the nth card is the first black ace is (copying the answer from Pratik):

50/52*49/51*48/50*47/49……(52-n)/(54-n) * 2/(53-n)

= 2*(52-n)/(51*52)

To max prob across n – minimise n
So I will bet on n=1

Hope you all enjoyed the puzzle!

 

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