Predictive Hacks

Cards Probability Question

The following example could be a good interview question, so maybe it is worth reading it :). Let’s consider the following scenario. I pick a random card from a 52-card deck and you have to guess it. You can ask me one of the following two questions about my hidden card:

  • Is it Red?
  • Is it King of Diamonds?

Which questions will you choose to ask me and why?

Let’s try to find out what are the differences between those two questions.

Calculate the Probability to Win

We need to calculate the Probability of guessing correctly the card for each question made.

Scenario 1: You asked me “Is it Red?”

The probability to be Red is 50% and the probability not to be Red is again 50%. Thus, no matter what my card is, the probability to Win before the question was 1/52 and after the question became 1/26. The table below represents the probabilities of each event, and the “Probability to Win” is the sum product of ” Probability of X Answer” and ” Probability to Win Given the X Answer

Answer OutcomeProbability of X AnswerProbability to Win Given the X AnswerProbability to Win
Yes0.5  1/26  1/26
No0.5  1/26

Hence, to sum up, in this case the probability to win is 1/26.

Scenario 2: You asked me “Is it King of Diamonds”

Following again the same logic, we represent the calculations in the table below.

Answer OutcomeProbability of X AnswerProbability to Win Given the X AnswerProbability to Win
Yes  1/521        1/26
No 51/52  1/51

As we can see, in this cases the probability to win is again 1/26!

Practically, the two question can be translated to “Do you prefer to have a guess on the half cards” or do “Do you prefer to have two attempts”. So what does this mean? Should we be indifferent to which question to ask since the probability to win is the same in both cases?

I do not have a clear answer on this. I think it is related to how you perceive the problem. Depending on how you will approach it, you come up with a different value of variance for the second question. I would be happy to see your answers!

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