Groups of size three are formed, blindfolded and then each member receives a hat

Question

Groups of size three are formed, blindfolded and then each member receives a hat that is either white or black and this will be done completely at random and independently. At a signal the blindfolds are removed and each person can see the other individual’s hats but not their own. The group members must simultaneously guess the color of their own hat or pass. No communication of any sort will be allowed at this stage. They will be set free only if there is at least one non-pass and there are no incorrect answers. What should they do to maximize their chances?

1. There is an obvious strategy with a 50% chance of winning. What is that?

2. At a strategy session the night before one mathematician comes up with the following: “Either all hats are the same colour or they are mixed, two of one or one of the other and the latter is the most probable. Thus if one of us sees two hats of opposite colours we can add nothing - we just pass. But one of us must see two hats of the same colour and to be in the set with larger probability should declare their hat is of the opposite colour.” Can you follow this thread and see if it indeed gives a better strategy?

Explanation / Answer

1)Each member can select one member who guess whether the hat is either white or black hat and the other two member can use the pass option.In this case there would be a 50% chance of winning.

2)Consider a case of single group where the possible combination of hat or the sample space is

{WWW,WWB,WBW,BWW,BBB,BBW,BWB,WBB}.

Strategy

The person which is going to guess should give the colour which is opposite to the colour of the two persons hat.This will work in 75%(6/8) of the case as there are 6 sample with two same colour.They would be wrong in 2 cases where the colour are same.

So the two members can pass and the other can guess the opposite colour so this always work.

It is a better strategy as the chance of winning goes to 75% from 50%.

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