DART Poster Magenta

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The Unfinished Game - There are many ways of discussing this.

- Shouldn't they just split the money as the game isnt completed?

- Isn't the leading hand likely to win, so shouldnt he get the money?

- Shouldn't they split it 5 : 2?

The problem is set out as originally expressed. But how about looking at it another way. Suppose the game was to continue and you were to offer to buy one of the hands? How much each they be worth?

That depends on the probability of either hand winning if the game continued.

Imagine the person on "heads" is on 5

and the person on "tails" is on 2

The next throw can be either heads or tails

The person on tails needs to win the next 4 throws in a row. Four tails is only one combination possible out of all the combinations possible over 4 throws.

There are two possible results for each toss (H or T) and for each of these on the following toss there are 2 more possibilities.

So after 4 tosses there should be 2* 2* 2*2 = 16

As only one of these possibilities will give Tails a win the probability for tails is 1/16 and for Heads to win it is 15/16.

So the money should be split in that order!!

The combinations are

HHHH

HHHT

HHTT

HHTH

HTHH

HTHT

HTTH

HTTT

THHH

THHT

THTH

THTT

TTHH

TTHT

TTTH

TTTT    Tails Wins!

 

 

 

 

See

The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern by Keith Devlin

The Monty Hall Conjecture

You should change. When you make your first choice, you have a 1/3 chance. Therefore there's a 2/3 chance that the car is behind one of the other doors. Therefore by changing you move from a 1/3 chance to a 2/3 chance. You wont necessarily win, but you double your chances.