The Painted Die

Let's play a simple dice game, no numbers - just the colors red and green.  We take a die and paint four of the faces green and 2 of the faces red.  The die is fair so any one of the 6 sides is equally likely to appear.  Now we roll the die 20 times and record the sequence of red and black events. But before the die is rolled 20 times you get to choose one of the following sequences:

Sequence 1: RGRRR
Sequence 2: GRGRRR
Sequence 3: GRRRRR

If your chosen sequence appears anywhere within the sequence of 20 events, you will receive $25.  So which sequence would you pick?

Researchers conducted this experiment with volunteers.  It turned out that 60% of the volunteers picked Sequence 2. Surprisingly, the scenario most likely to occur is actually Sequence 1!  Why?

Look carefully at Sequence 1 and you will notice that it is a subset of Sequence 2. I.e. anytime Sequence 2 occurs, Sequence 1 will ALWAYS occur. But when Sequence 1 occurs, Sequence 2 may or may not occur.

We are drawn to Sequence 2 since it contains 2 green results whereas the other sequences contain just 1 green result. And since green is more likely than red, we choose Sequence 2.

Our intuition which guides us in determining the perceived likelihood of the events does not misleads us. Once we learn the basics of probability we will be better equipped to challenge our intuitive assumptions and be better able to determine the correct results.