Winning Marble
How can this prisoner escape?
Age
14 to 16
Challenge level
A prisoner is given a chance to win her freedom.
She is given two boxes and some marbles. She must put the marbles into the boxes.
Afterwards, she will be blindfolded, and will choose a box and select a marble. If she draws a white marble she will go free; otherwise, she will remain in prison.
The prisoner has 10 white marbles and 10 black marbles. She can put as many of the marbles of each colour as she likes into each box. She must use up all the marbles.
How can she maximise the probability that she will go free? In that case, what is the probability that she will go free?
This problem is taken from the World Mathematics Championships
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Probability of getting a white marble:
$\frac12\times(${proportion white in box 1} $+$ {proportion white in box 2}$)$
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If half the marbles in each box are white,
the probability that she wins her freedom is $\frac12$
Otherwise, one box will be more than $\frac12$ white and the other will be less than $\frac12$ white.
To maximise her probability of freedom we need to get as close to $1$ in one box and to $\frac12$ in the other box as possible. We do this by putting just one white marble in one box:
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Probability of freedom is $\frac12\times(1+\frac9{19})=\frac{28}{38}$, which is nearly $74\%$.