Problem 121

[LarryC]

Hi

Problem 121 looks easy but I am unable to verify that the probability of winning is 11/120 as stated in the question. I don't make it that at all and am pretty sure of my answer so it would be useful to see if my earlier answers agree with what is expected.

Can anyone verify:
Rounds | Probability Of A Win
1 | [frac]1,2[/frac]
2 | [frac]1,6[/frac]
3 | [frac]5,18[/frac]
4 | [frac]1,9[/frac]

That would be much appreciated...
Thanks,
Lster

[Tommy137]

5/18 for 3 rounds is not correct.

[LarryC]

Thanks! Are the first two rounds correct?

[stijn263]

Yes they are.

Remember that a player can win a game of three rounds if he takes:
- 3 blue disks (bbb)
- 2 blue disks and 1 red disk (bbr, brb, rbb)

[LarryC]

After each turn the disc is returned to the bag, an extra red disc is added, and another disc is taken at random.

I've just realized I may well be interpreting this in a way that is not intended.

Does the above statement imply that the second disk taking is also "noted" or not? It could be interpreted either way and my results assume that the disk is only removed.

[Tommy137]

Example:

bag contains: B , R
player takes B

now another red disc is added

bag contains: B , R, R
player takes: R

another red disc is added

bag contains: B, R, R, R
player takes: B

=> The player took 2 blue discs and 1 red, so he has won. (probability 1/2*2/3*1/4 = 1/12)

[LarryC]

Got it! Thanks for your help guys.

After each turn the disc is returned to the bag, an extra red disc is added, and another disc is taken at random.

This sentence is very ambiguous. It's primary meaning seems to imply that a disc is randomly removed after the turn and before the next. An example, similar to Tommy137's one, would go a long way! Or perhaps it should be rephrased.

[Tommy137]

Got it! Thanks for your help guys.

Gratz

After each turn the disc is returned to the bag, an extra red disc is added, and another disc is taken at random.

This sentence is very ambiguous. It's primary meaning seems to imply that a disc is randomly removed after the turn and before the next. An example, similar to Tommy137's one, would go a long way! Or perhaps it should be rephrased.

This interpretation did never occur to me, but people may understand it that way.

[ImRe]

After reading this topic, I still don't fully understand this problem.
How are the payouts done here? In the four turn game, is the player getting back £1 for every round (s)he won, or for every four turn (s)he won, or is it something completely different?

Thx

[Tommy137]

The payout's done after the complete game is played. If the player wins the four turn game, he gets £10 including his stake of £1.

[Knut.Angstrom]

How much is the player paid if he wins after n turns? After 4 turns he is paid £10 so much I understand

[Tommy137]

Please specify your question... and you could read the already existing topic (viewtopic.php?f=50&t=997) for Problem 121 and see if it's helping you.

[rayfil]

How much is the player paid if he wins after n turns?

If you read the problem description very carefully, THAT is the answer you have to find.

Statisticly, it would be the largest payout by the sponsor of the game such that no overall loss (and minimal gain) would be expected in the long run. (If the payout would be £11 when the probability of winning is 11/120, £121 would be given for every £120 of revenue.)

[estanford]

This problem confuses me. How much of the prize fund does the player win as a consequence of a single win event? Does it vary according to some function or is it a constant?

[TripleM]

I'm not quite sure what you mean by 'single win event'; there is only one outcome. If the player has won after 15 rounds, they receive the whole prize.

[estanford]

So am I understanding the question correctly if I read it like this?

~~~~~~~

Let there exist three piles of money:
1) The money the player has (assume infinite),
2) The money the player has given as entry fees,
3) The prize money.

Each time the player plays the game, they pay a fee of 1 pound. This money is moved from pile (1) to pile (2). Money is never transferred from pile (1) or (2) to pile (3). The size of pile (3) is an integer constant, set by the banker at the outset of the game. The player keeps playing until s/he wins. Given that the game is played for n turns, what is the smallest possible size of pile (3) such that the banker does not expect to lose money by the time the player wins?

~~~~~~

EDIT: After solving the problem, I see that the answer was 'yes'. Good times.

[chiefsci]

One quick question. I thought that I had calculated the answer with a pen and paper, but it is not being accepted. Am I correct in using combinatorics (like those in Problem 53) to calculate the numerator of the fractional form of the probability of success after 15 rounds? It works perfectly for the 4 round example, producing 11 for the 11/120, but my current answer isn't being accepted.

[GenePeer]

I did it by Paper/Pencil too but the calculations involved never required to find any binomial coefficient! The four turns example isn't enough, work on 5-7 turns to properly understand the question, for 7 turns the payout is 17. Enjoy.

[Francky]

I can find the value for n=2000 in 1.3s

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[enderw88]

Seems like the 4 turn example considers drawing 2 Blues and 2 Reds as a win. Or am I reading it incorrectly?