Coins Puzzle: A Simple Problem With a Counterintuitive Solution

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Introduction

I found this problem as part of the Haselbauer-Dickheiser Test. It is a very simple and elegant problem with a counterintuitive solution.

Problem

A box contains two coins. One coin is heads on both sides and the other is heads on one side and tails on the other. One coin is selected from the box at random and the face of one side is observed. If the face is heads, what is the probability that the other side is also heads?

Solution

The first instinct many people have is to answer 12=0.5=50% because the unobserved side is either heads or tails, so 1 out of 2 will be heads. However, the reason this is not the correct answer is because those two possibilities don't have the same probablity of occuring.

Let h represent heads and t represent tails. The coin with heads on both sides will be represented as hh and the coins with heads on one side and tails on the other side will be represented as ht. Futhermore, let's use an uppercase letter instead of a lowercase one to represent the side we observe.

So there are obviously 2 possibilities for the selected coin: hh and ht. However, after observing one side, we are left with 4 possibilities with equal probability each:

  1. Hh
  2. hH
  3. Ht
  4. hT

However, we know that the observed side was heads, so we can discard option 4. We are thus left with 3 possibilities with equal probability each:

  1. Hh
  2. hH
  3. Ht

In 2 out of the 3 possibilities here, the unobserved side is heads. Thus, the answer is 23≈0.67≈67%.