perplexus.info :: Probability : The King Shows Some Mercy

The King Shows Some Mercy (Posted on 2020-10-19)

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.

Submitted by Charlie

Rating: 3.0000 (2 votes)

Solution: (Hide)

This table shows the probability all four of you're draws will be red (taken from a larger table with smaller probabilities) given the number of reds in each vase:

3 3 5 5 0.3600 3 4 4 4 0.3072 3 4 4 5 0.3840 3 4 5 5 0.4800 3 5 5 5 0.6000 4 4 4 4 0.4096 4 4 4 5 0.5120 4 4 5 5 0.6400 4 5 5 5 0.8000 5 5 5 5 1.0000

The row with three vases with 4 red balls each and one vase with all 5 balls red is the closest to probability of 1/2, thus giving the king the lowest probability of being right; though that's 51.2%, it's the best you can do.

for u1=0:5 u(1)=u1; for u2=u1:5 u(2)=u2; for u3=u2:5 u(3)=u3; for u4=u3:5 u(4)=u4; p=1; for i=1:4 p=p*u(i)/5; end fprintf ('%3d %3d %3d %3d %8.4f \\n',u(1),u(2),u(3),u(4),p) end end end end

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(2): If I understand the problem	Jer	2020-10-20 08:12:43
re: If I understand the problem	Steve Herman	2020-10-19 10:55:24
Solution	Steve Herman	2020-10-19 10:53:16
If I understand the problem	Jer	2020-10-19 10:53:00

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