Earlier today I asked you the following puzzle, about voters who give wrong answers to opinion polls because they embarrassed to admit to their preferences. Here it is again with a solution.
The shy voter puzzle
Imagine an election with two candidates, say Trump and Harris. Many voters are embarrassed to admit who they vote for.
You are a pollster, going from door to door. You have a coin in your pocket. Your aim is to find the percentages who will vote for each candidate.
Can you think of a polling method that makes voters comfortable to give their honest preference, even if they are embarrassed to admit their preferred candidate to you?
The method must result in a pretty accurate poll, although it might not be 100 per cent accurate.
Solution It’s all about the coin.
Ask each voter to flip the coin twice in private. There are four equally likely outcomes: TH, HT, TT, HH.
(Am I the first person to notice that in this toss-up election, one of the candidates is T and the other is H?)
Tell the voter the following: if you flip TT, say that you will vote for Trump. If it is HH, say Harris, and if the outcome is TH or HT, give your honest preference.
This method gives the voter plausible deniability. If the voter responds that they will vote for Trump, either they flipped TT or if they are responding honestly, and the pollster does not know which. Likewise If the voter responds that they will vote for Harris, either they flipped HH or if they are responding honestly, and the pollster does not know which. If you are a shy voter, you will be less embarrassed in giving your true preference because you can always deny that what you told the pollster is your preference – it could just be the coins!
If you ask all voters to flip the coins in this way, at the end of the process, about 25 per cent will have flipped TT and about 25 per cent HH. So, if the final tallies are X per cent for Trump and Y per cent for Harris, you subtract the ‘coin’ votes from the real ones to get the final total. The actual ratio of Trump to Harris voters is X-25/Y-25.
The method is a technique – called ‘randomised response’ – that has been used since the 1960s for surveys about sensitive issues, such as drug use or sexual preference, although I am not aware that pollsters do this for presidential or general elections. If any pollsters are reading this, perhaps they can let us know in the comments below how common the method is, or if it has been used in a political context?
I hope you enjoyed the puzzle, I will be back in two weeks.
Think Twice: Solve the Simple Puzzles (Almost) Everyone Gets Wrong. To support the Guardian and Observer, order your copy at guardianbookshop.com. Delivery charges may apply. (In the US, the book is called Puzzle Me Twice.)
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.