A Probability Conundrum

The conundrum

You and I are sitting opposite each other across a table.

I have four cards.  Two of them have red circles on their fronts and two have black circles.  The backs of all the cards are identical.  I shuffle them without looking at them, and place them face down on the table in front of me.



What is the probability that circle on the second card is red?

I now pick up the top card and look at its front without showing you.

Now what is the probability that the circle on the second card is red?  How would I answer this question, and how would you?

(You might find it helpful to draw a tree diagram or to list all the possible orderings of the four cards.)


A similar question could equally well be asked about a regular pack of cards, or any other situation in which different people know different things about the world.

After thinking about this conundrum and reading the discussion below, you may want to explore or revisit the problems in the Probability and Evidence collection, and consider what we mean by the term "probability" in each case.

Please have a think about the questions above before reading further.

Discussion

Initially, as there are two red and two black cards, the second card is equally likely to be red or black, so the probability that it is black is $\frac{1}{2}$.

Once I have looked at the top card, things change.

If the top card was black, I know that there are 1 black and 2 red cards left, so the probability that the second card is black is $\frac{1}{3}$.  But if the top card was red, I know that there are 2 black and 1 red card left, so the probability that the second card is black is $\frac{2}{3}$.

On the other hand, you don't know anything about the top card, so as far as you are concerned, the probability that the second card is black is still $\frac{1}{2}$.

This is strange: you and I calculate different probabilities for the same event!  How can this be?

Perhaps we might say that this is not fair: we have different amounts of information about the world.  Maybe you would suggest that we shouldn't be using probabilities at all - after all, the second card has a definite colour, we just don't yet know what it is.  It's not like rolling a die or tossing a coin, where the action hasn't yet happened.  However, it is common to use the language of probability in situations like this.  If you wanted to make a bet on the colour of the next card, you could use probability theory to decide how to bet, but you would not take my knowledge into account, as you do not know it.  (Though you would probably not want to be against me!)

Another example is that the fire service might want to know the probability that a certain forest fire was caused by lightning as opposed to by arson.  The fire either was or was not caused by lightning, so how can probability be useful here?  Nevertheless, it appears that in this situation we are using probability to describe our confidence or degree of belief in something, and that would apply to the red and black cards question as well.  Maybe that is what probability "is"?

But others would suggest differently.  For example, we learn that the probability of a coin toss landing heads is $\frac{1}{2}$, and we learn to calculate the probability of a pair of coins landing as two heads, and so on.  Are these based on our degree of belief?

If we think about the National Lottery, there are many people who always bet on the same numbers, as they believe them to be their lucky numbers; they presumably believe that these numbers are more likely to come up than other numbers.  Yet using "theoretical probability", we can say that the probability of any particular set of six numbers coming up in the lottery is the same as that of any other set of six numbers.  So these people are acting on the basis of their degree of belief, even though the theoretical probability doesn't match it.

This is a difficult question, and there are no simple answers.  This question pervades the whole of statistics, and leads to very different approaches to inferring information from data.  If you would like to explore this topic further, you can find a thorough discussion in the Stanford Encyclopedia of Philosophy, or look up the terms "aleatory uncertainty" (the world is random and we don't know what will happen in the future) and "epistemic uncertainty" (we are just ignorant of what the reality is or will be).


Some notes for teachers on the use of this article can be found here.

This resource is part of the collection Statistics - Maths of Real Life