A Probability Conundrum

Age 14 to 18
Article by Julian Gilbey

Published October 2018.

Why think about the problem in this article?

This is a useful problem for deeping students' understanding of probability.  "What is probability?" is a hard question, and this problem is designed to elicit this question and open it up for discussion.  There is no "correct" answer – rather, it is a subject of significant debate.  There is a discipline known as the Philosophy of Mathematics, and this question is an important one in that field.

Possible approaches

The article could be given "as is" to students to read and discuss.

Alternatively, the question in the article can be posed to the class as given, it could be reduced to the case of just two cards (one red and one black), or it can be given in the context of a normal deck of 52 playing cards.  Having just two cards might cause more confusion, as the person looking at the card will then be certain about the colour of the remaining card.  If desired, it could be acted out with the teacher and a student in front of the class to make it more vivid.

If students do not ask a question along the lines of "What is probability, anyway?", you could encourage them by asking "How can you and I have different answers to this question and both be correct?"

Key question

  • What do we mean by "probability"?
This is a difficult question, and there are no simple answers.  If you would like to explore it further, you can find a thorough discussion in the Stanford Encyclopedia of Philosophy.

Possible extension

  • Are there situations when everyone would agree on the probability of a certain event happening?
  • Can you think of other situations where different people would give different probabilities for something?

Possible support

Drawing tree diagrams to represent the situation may be helpful for some students, as might writing out a list of all 6 possible orderings of the four cards.