Sally Sugar had twelve sweets, of which there were four of three different types: toffees, orange creams and nutty chews. Her two friends, Pierre and Gael, each took a sweet, leaving Sally with ten sweets to enjoy herself.
Is there a now a greater or less than 50% chance that Sally still has all four of her favourite sweet, the toffee, left to enjoy?
This is a ‘Logical Maths Teaser’ from a book of ‘brain games’ that my daughter gave me to help pass the time while travelling. But it does seem similar to some real-life questions that I have got involved in. Maybe you would like to think about it and even debate possible answers?
The official answer
The answer in the back of the book is:
less than 50% chance
This should be no surprise to anyone familiar with such puzzles. But can we justify it? (Have a go!)
Justifying the ‘official’ answer
My best attempt to say that this is a popular book, and as such the objective – as in an exam – is not to determine the actual probability but what the authors think it is. In this case, ‘the rules of the game’ seem to be that you should apply probability theory without thinking too deeply. But even here there is a problem:
- If Pierre and Gael choose a sweet at random then the chance of neither taking a toffee is (8/12).(7/11) = 14/33 < 1/2.
- If Pierre and Gael choose their favourites, the probabilities will be different. We do not know what there favourites are. According to the principle of indifference, we can suppose that there is a probability of 1/3 that they prefer toffee, hence a probability of (2/3).(2/3) = 4/9 <1/2 that they do not prefer toffee and hence that Sally has four left.
So far, it does seem that the answer provided is correct, but troubling that the probability seems under-determined. Suppose now that, out of the flavours available, the proportion that prefer toffee is ρ. Then (with the usual assumptions) the probability that Sally has four toffees left is (1-ρ).(1-ρ), hence if the answer is correct then no more than 30% of people prefer toffees. But is this a valid application of the principle of indifference?
My own view
Pierre and Gael are friends, and so – probably – would know that Sally preferred the toffees, and would leave them for her. It even seems more likely to me that Sally had orange creams and nutty chews because her friends preferred them than that her friends would take her favourites.
I may be wrong in my probability estimates, but I do not see that I am wrong in my ‘logic and maths’. To me, reasonable answers would include ‘don’t know’ and a discussion of what the answer would be for various assumptions, focussing – as I have done – on the differences that make a difference to the outcome, and a search for data to help resolve these issues.
If we had not been told that the three were friends or that the sweets were in three types, but only that Sally had four favourite sweets and eight others, then the answer provided might be more reasonable. If we were then given the additional information then advocates of Baye’s rule would suggest updating the probability estimate accordingly. But what does
P(“The three are friends” | “Sally has her four favourite sweets” )
mean? It seems to me that Bayes’ rule is not as universal as it is sometimes held to be.