Can you save the astrologer ?

Probability Level 2

Once upon a time there was dictator. An astrologer forecast something bad for him; so the dictator awarded death penalty to the astrologer. The latter pleded for his life, so the dictator gave him a chance to save himself and decreed as follows: "I will allow you to put 2 white balls and 2 black balls in any manner you like in two urns without disclosing it to anybody. My executioner will choose one of the urns, dip his hand into it and take out a ball. If the bll is black he will cut of your head and if its white, then you are saved." What would you advise the astrologer to do, in order to give himself the maximum probability of saving his life $?$

no matter what he does place 1 white ball in urn1 and 1 white and 2 black in urn2 place 2 white ball in urn1 and 2 black in urn2 place 1 white ball and 1 black in urn1 and 2 black in urn2

1 solution

Marco Brezzi
Aug 8, 2017

$\textbf{Case 1:}$ two balls in every urn

We can have $BB - WW$ which gives a probability of $\dfrac{1}{2}(0+1)=0.5$ or $BW - BW$ which also gives $\dfrac{1}{2}\left(\dfrac{1}{2}+\dfrac{1}{2}\right)=0.5$

$\textbf{Case 2:}$ three balls in one urn and one in the other

Again two subcases: $B - BWW$ which gives $\dfrac{1}{2}\left(0+\dfrac{2}{3}\right)=\dfrac{1}{3}$ or $W - WBB$ with a probability of $\dfrac{1}{2}\left(1+\dfrac{1}{3}\right)=\dfrac{2}{3}$

We can see that the case with the higher probability of surviving is $\boxed{W - WBB}$

Can you save the astrologer ?

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