You're imprisoned by a sadistic king, who's also rather eccentric. He's devised a game in which he will guess whether a ball of a given color will be retrieved when you retrieve by randomly selecting one ball from each of four opaque vases that you will fill in his view before the selection process.
There's a sufficient supply of both red and white balls. The rules are that you will put five balls in each of the four vases. You can choose how many of each of the two colors, but the king sees how many of each you use in each. You will then select one ball from each of the four vases. The king will guess Yes or No that at least one of the balls will be white. If he's right you'll stay imprisoned or worse. If he's wrong you'll go free.
What is the best strategy for maximizing your probability of gaining your freedom? That is, what allocation of colors should be made for each vase; they need not be the same for all the vases, but each must have a total of five.
5 urns. The kings gets to guess whether "at least one of the balls is white."
Presumably we want the probability to be close to 1/2 since if it isn't the king will choose whichever is more likely.
The inverse of "at least one white" is "all red" which is easier to calculate.
Let the numbers of red balls in the four urns be notated {a,b,c,d}.
The probability of all red is then (a/5)(b/5)(c/5)(d/5) = (abcd)/625.
To be close to 1/2, abcd must be close to 625/2=312.5
Playing around with different possibilities I found the closest is {4,4,4,5}=320, the next closest is {3,4,5,5}=300
Solution:
place 4 red and 1 white into three of the urns, and 5 red in the fourth urn.
P(at least one white) = 1-P(all red) = 1-320/625 = 1-.512 = .488
The king will guess NO and have a 0.512 chance of being correct.
You have a 0.488 chance of freedom.
|
|
Posted by Jer
on 2020-10-19 10:53:00 |
If I understand the problem
