## January 9, 2009

### The six dice problem

Each has been rolled out of your sight and hidden behind a cup.
You are informed that no more than one of them landed a 6 - but
it may be that none of them did.

You are going to remove the cups one at a time.

Before you do, you label each cup with letters A to F.
This designates the order in which you will remove them.
You have complete discretion which cup is assigned each letter,
but once labeled they must be removed in alphabetical sequence.

Having labelled them, it is time to remove the cups accordingly from A to F.

If one of the dice landed as 6, under which cup is it most likely to be found?

Secret Squïrrel said...

Before any cups have been lifted it will be equally as likely to find a six under any of them, since they were labelled blindly.

As each cup is lifted and a six not found, the chance if it being under any particular cup increases but it always the same for each remaining cup.

Some people incorrectly extrapolate this to mean that since the probability increases as you remove cups, then the last cup has the greatest total probability. However, they are neglecting to factor in the cases where a six is found beforehand. For example, if its found under the first cup then the probability of finding it under the other 5 cups is zero. Add the probabilities for ALL of the possible cases and it works out to the same for each cup - 1 in 6 or approx 0.167.

It is easy to demonstrate this by actually performing the experiment.

Evan said...

16.7%? Dont' be Rediculous! There are only 3 relevent sentences in the question:

1. There are six unloaded die.
2. Each has been rolled out of your sight and hidden behind a cup.
3. If one of the dice landed as 6, under which cup is it most likely to be found?

The answer is... None of them!

READ the problem. Each die has been hidden "behind a cup". None of the dice will be found under the cup! So silly!

Secret Squïrrel said...

Evan, now you are just being annoying. Again you twist words to fulfil some perverse urge of yours to be a right knob.

If you had a more complete understanding of English and mathematics, you would know that the concept of "behind" is relative to the coordinate system in use and is not restricted to the xy plane as you suppose.

Anonymous said...

this puzzle is pretty random,I dont understand chance.

tamal said...

what if we assume that the probability density is 1/6 as you pointed out and consider the mean to be most likely i.e 1.1/6+2.1/6+....+6.1/6=3 rd cup.

Jürgen said...

To: Secret Squirrel

Why would the chance to find a 6 increase, when u have opened a cup?

When you haven't opened any of the cups the highest chance to find a die with a number of 6 in a cup, is when all cups are closed.

Same with the lottery, you have higher chance to win when you buy more tickets.