[superprismatic]

Here is my modification of a puzzle by the late

great puzzle maker, Walter Penney:

There are 309 white balls and 191 black balls in

a bag. A man makes a deal in which he draws balls

one at a time from the bag, replacing and mixing

after each draw, and if the first black ball is

drawn on the Nth trial, he is to receive a number

of dollars equal to the Nth Fibonacci number

(0,1,1,2,3,5,8,13,... for N=1,2,3,4,5,6,7,8,...).

Two questions:

1. What is the expected payoff for the man?

2. If he were to surreptitiously place one more

white ball in the bag, by how much does it improve

his expectation from that in question 1?

[Guest]

I like this one

1. $6946

2. Infinite

I took the Fibonacci sequence as a geometric series. For part 1 the ratio (black to white) is less than phi but very close. However, for part 2 the ratio is greater than phi which causes the series to diverge.

[Guest]

well psychic mind is normally pretty good so I might be wrong but this is what I got

well I cant tell you how wrong i was when i started typing this i read it as 309 balls and 191 of them were black which coincidentally just reverses the probabilities and I started typing this saying you had it backwards (the type above that is also from that but i still think the answer is a lil off

I got 5026.32.....

I did this by the summation (E)

E=wE+w^2E+1

E-wE-w^2E=1

E=1/(1-w-w^2)

E=13157.89474

but then i never did the black ball so multiply by chance of Black

E=5026.325789

I completely agree about part 2 tho

and it really was a good problem It was interesting

[superprismatic]

I don't agree with your part 1 and I can't seem to reproduce how you got it. But, I'm sure it's a small matter. Congrats on getting part 2 spot on. I just think it's pretty cool that a single ball makes all that difference!

You'r part 1 answer was for the problem if the Fibonacci started 1,1,2,3,5,.... instead of 0,1,1,2,3,5..... You had a neat, simple way of doing it, too.

[Guest]

I have made the same mistake. Well I did this again quickly.

4292.68

[Guest]

ok guess number two.. i mean definitely the answer number 2

so i relooked at it and the mistake i missed was 1wb at the beggining

so

summation E prob white W prob black B

E=WE+W^2E+W

E=W/(1-W-W^2)=8131.....

but that is forgetting the B

so E=8131.578947*B=3106.263158

Im pretty sure this is right now

anyway I tried to look up this problem you were talking about. can you tell me what this problem is or what website i could find it on.

[superprismatic]

ok guess number two.. i mean definitely the answer number 2

so i relooked at it and the mistake i missed was 1wb at the beggining

so

summation E prob white W prob black B

E=WE+W^2E+W

E=W/(1-W-W^2)=8131.....

but that is forgetting the B

so E=8131.578947*B=3106.263158

Im pretty sure this is right now

anyway I tried to look up this problem you were talking about. can you tell me what this problem is or what website i could find it on.

Spot on! Part 1 is indeed that!

I can't give you a website because I got it from some privately published

papers containing original Walter Penney problems. I'm glad you liked this

one -- I'll put out more from time to time. I won't do it too quickly, though.

Walter Penney was a master puzzle maker and I don't want to monopolize

things. Walter died in 2000 at the age of 87.

Edited July 26, 2009 by superprismatic