I would like to introduce a probabilistic way of reasoning by presenting a few proposition bets and analysing them. In proposition bets by con artists, the bet appears to be advantageous to the person being presented with the bet. That person is usually referred to as “the sucker” or “mark” by the con man. Hence, these types of bets are also referred to as sucker bets.

Here is a sucker bet: The con man presents you with three cards A, B, and C. Card A, has the colour blue on both sides. Card B, has the colour red on both sides. Card C, has blue on one side and red on the other. The con man asks you to shuffle the three cards and then asks you to place the three cards inside a box. You both leave the room, and a third person picks one of the three cards from the box and places it on the table. Both you and the con man return to the room. You see that the top colour of the card on the table is red. Pointing at the card on the table, the con man says to you and bets that the other side of the card is red. Should you take the bet?

Bogus Solution: What's wrong in this "solution" here?

The card on the table cannot be Card A so we can eliminate it. Therefore, the card on the table is either Card B or Card C. If the card is card B we lose as the other side will be red. If the card is card C the other side will be blue and so we win. Therefore, the odds of you winning the bet are 50-50.

Mistake: We forgot that card B has two red sides and hence we actually get two separate cases, both of which we lose. Here is the complete solution:

Solution: The card on the table cannot be Card A so we can eliminate it. Therefore, the card on the table is either Card B or Card C. Let the B card have two faces, namely 1 and 2. If the card is the C card, then the other side must be blue and you win. If the card is card B with face 1 facing upward, you lose. You also lose if the card is card B with face 2 facing upwards. Thus we win in only one case and lose in two cases. Therefore, you must not take the bet.

We must not jump to conclusions while analysing such problems but carefully look into all possibilities. Such is the case here as most people forget that there are three possible red faces and not two and hence feel that there is a 50-50 chance of winning the bet.

Other than learning to analyse, having fun is also extremely important. Don't read the blog as you would read a textbook but rather read it the way you would read a novel. Read with the intention to explore. Have fun playing the bet on your friends, teachers and parents and see if they take the bet.

Author: Sriteja, Class 10, Meridian School, Madhapur, Hyderabad.