Question

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 marbles of each color as she likes into each box. She must use up all the marbles. How can she maximize the probability that she will go free? In that case, what is the probability that she will go free?

Answer 1

The prisoner can maximize her probability of drawing a white marble by placing one white marble in one box and the rest of the marbles in the other box, resulting in a probability of 14/19 of drawing a white marble and gaining her freedom.

Step-by-step explanation:

To maximize the probability of drawing a white marble and winning her freedom, the prisoner should use a strategy that involves an unequal distribution of marbles. She should place one white marble in one box and all the remaining marbles, which includes ten black and nine white marbles, in the other box. When she is blindfolded, she has a 50% chance of picking either box. If she picks the box with only one marble, her chance of drawing a white marble is 100%. If she picks the other box, her chance is 9/19 since there are 9 white marbles out of 19 total marbles.

The overall probability (P) can be calculated by adding the probabilities of both scenarios:

P = (1/2 * 1) + (1/2 * 9/19) = 1/2 + 9/38 = 19/38 + 9/38 = 28/38, which simplifies to 14/19.

So, by using this strategy, the prisoner maximizes her chances, and the overall probability of her winning her freedom is 14/19.