What do you think?

All_Is_Number · 2011-01-25 17:45:05

bobbym wrote:

Hi All_Is_Number;
Thanks for looking at the problem but there is definitely if I remember an answer for every part of it. The problem answer is not posted here because It was being contested on another forum ( now defunct ). Since they were aware of me here I did not want them to just click a hide button.

I'm reasonably confident more information is needed. In particular, I believe the number of bets actually laid, and the probability of a success for each particular bet offered (information not available from the expected value alone) is needed.

I would be interested to see your solution, including how you worked it out. I've been wrong before, and the probability of me being wrong again at least once at some point in time is 0.999999. :-D

Howardroark · 2011-01-25 17:48:28

Hi bobbym,

bobbym wrote:

A real killer!
A man walks through a tunnel. Two thirds of the way through he sees a car heading for him. The car is moving at 75 mph. If he starts to run he can get to either end of the tunnel just as the car does. What is the speed in mph of the running man?
A says) Impossible! You have to know the distance the car is from the man.
B says) I don't think so, but I am unable to solve it.
C says) I solved it. It was easy. The man is moving at 25 mph.
D says) I agree with B. It can be solved but C's answer is wrong. But I can't prove it.
What do you think?

Last edited by Howardroark (2011-01-25 17:48:46)

bobbym · 2011-01-25 17:48:39

Hi All_Is_Number;

I am working on the solution now as I cannot find it. But please work on your answer as it is not correct.

Just saw your answer Howard it is correct! very good!

Howardroark · 2011-01-25 17:55:14

Hi bobbym,

bobbym wrote:

Super easy!
Prove that:
Without using any particular value of sin(x) or cos(x).

Howardroark · 2011-01-25 18:04:36

Hi bobbym,

bobbym wrote:

Who eats the pie?
A,B,C,D and E decide to stop arguing and to do something together for a change. B's Mom bakes them a big pie to celebrate their togetherness. Before they start to eat they begin to argue about primes. They are mathematically inclined after all. They settle down on this one particular question.
If we start with the set of all fractions with 1 as the numerator and all the primes as denominators ( { 1 / 2, 1 / 3, 1 / 5, 1 / 7, 1 / 11, ...} and each of us starting with A and in order ( A,B,C,D,E, A,B,C,D,E... ) takes the next fraction and eats that amount of the remaining pie. In other words A eats 1 / 2, B eats 1 / 3 of the remaining half, C eats 1 / 5 of the remaining sixth... When we get to E and if there is more pie left we start again with A. The question is will we ever finish the pie?
A says) Since this can go on forever I think there will always be some pie left.
B says) Not necessarily sometimes an infinite process can equal a finite number. I think the pie will eventually be consumed.
C says) Who cares I am hungry.
D says) No wait, I think I read about something like this called Zeno's paradox.
E says) Hey D, did you make that up?

All_Is_Number · 2011-01-25 18:05:48

bobbym wrote:

Hi All_Is_Number;
I am working on the solution now as I cannot find it. But please work on your answer as it is not correct.

If, by "players expectation is 18 / 37 dollars per game" you mean expected value of each bet, and not expected gain on each bet, my answer is correct.

If you meant expected gain, my methodology is incorrect. If you meant the probability of a player winning a particular bet is 18/37, then my methodology is incorrect.

Last edited by All_Is_Number (2011-01-25 18:06:46)

bobbym · 2011-01-25 18:09:12

Hi Howardroark;

Yes, that one is an insult. But your answer was better than mine. So very good work.

bobbym · 2011-01-25 18:18:56

Hi All_Is_Number;

I will change the problem to make it less ambiguous.

3) A guy owns a small casino. It only has one gaming table. He would like to earn 1000 dollars a day with it. There is only a flat bet of 2 dollars per roll and the players probability is 18 / 37 of winning that bet. How many games must be played on average for him to earn his 1000 dollars. Since he doesn't like to lose he needs to also know what is his chance of losing in any single day.

That clears up everything. I will adjust the original post when the smoke clears.

Hi Howardroark;

I think so too but I cannot prove that!

All_Is_Number · 2011-01-25 18:24:10

bobbym wrote:

Hi All_Is_Number;
I will change the problem to make it less ambiguous.
3) A guy owns a small casino. It only has one gaming table. He would like to earn 1000 dollars a day with it. There is only a flat bet of 2 dollars per roll and the players probability is 18 / 37 of winning that bet. How many games must be played on average for him to earn his 1000 dollars. Since he doesn't like to lose he needs to also know what is his chance of losing in any single day.
That clears up everything. I will adjust the original post when the smoke clears.

Not to be nit-picky, but what is the payout of the bet? Two dollars? A flat bet of $2 implies only that every bet costs $2 to place. It does not imply a particular amount paid out if the bet is won.

Last edited by All_Is_Number (2011-01-25 18:28:25)

bobbym · 2011-01-25 18:27:14

Yes, you are paid even money.

Howardroark · 2011-01-25 18:27:43

Hi bobbym,

Okay.. Thanks anyways..

bobbym · 2011-01-25 18:29:10

Hi Howardroark;

Sorry, that was one I put there at the request of someone else. I do not have an answer for it. I do not think the problem is particularly well posed.

All_Is_Number · 2011-01-25 18:29:17

bobbym wrote:

Yes, you are paid even money.

Howardroark · 2011-01-25 18:38:02

Hi bobbym,

No need of sorry bobby.. I need to thank you for posting wonderful probs.. Bobby, How about a starting a new topic on out of box thinking puzzles..?

bobbym · 2011-01-25 18:40:26

Hi All_Is_Number;

Yes, that is correct!

Now for the second part of the problem? What can I say to the casino manager about his chances of losing on any single day?

All_Is_Number · 2011-01-25 19:29:13

bobbym wrote:

What can I say to the casino manager about his chances of losing on any single day?

Assuming

games are played each day, the probability of the casino losing money on a particular day is

Last edited by All_Is_Number (2011-01-25 19:40:26)

bobbym · 2011-01-25 19:40:35

Hi All_Is_Number;

Excellent that is correct!

P(having a losing day ) = 0.0001212752342855117

3.677416458992032 standard deviations

He has approximately 1 chance in 8245 of having a losing day.

All_Is_Number · 2011-01-25 19:42:42

bobbym wrote:

Hi All_Is_Number;
Excellent that is correct!
P(having a losing day ) = 0.0001212752342855117

3.677416458992032 standard deviations
He has approximately 1 chance in 8245 of having a losing day.

How did you obtain such precision with P(losing day)?

bobbym · 2011-01-25 19:52:36

Hi;

That is not very precise at all because I numerically integrated the Normal curve with mean = 0 and SD = 1.

The actual answers are much closer to

P(having a losing day ) = 0.000117804.

He has approximately 1 chance in 8288 of having a losing day.

All_Is_Number · 2011-01-25 19:58:33

D'oh! I forgot all about integrating over the normal curve! That would have been easy enough to do.

bobbym · 2011-01-25 21:39:46

Howardroark wrote:

Bobby, How about a starting a new topic on out of box thinking puzzles..?

If they are really out of box thinking, then how the heck am I going to solve them?

Everyone did a good job today on the problems. Excellent work!

gAr · 2011-01-25 23:00:59

Hi bobbym,

Who eats the pie?
A,B,C,D and E decide to stop arguing and to do something together for a change. B's Mom bakes them a big pie to celebrate their togetherness. Before they start to eat they begin to argue about primes. They are mathematically inclined after all. They settle down on this one particular question.
If we start with the set of all fractions with 1 as the numerator and all the primes as denominators ( { 1 / 2, 1 / 3, 1 / 5, 1 / 7, 1 / 11, ...} and each of us starting with A and in order ( A,B,C,D,E, A,B,C,D,E... ) takes the next fraction and eats that amount of the remaining pie. In other words A eats 1 / 2, B eats 1 / 3 of the remaining half, C eats 1 / 5 of the remaining sixth... When we get to E and if there is more pie left we start again with A. The question is will we ever finish the pie?
A says) Since this can go on forever I think there will always be some pie left.
B says) Not necessarily sometimes an infinite process can equal a finite number. I think the pie will eventually be consumed.
C says) Who cares I am hungry.
D says) No wait, I think I read about something like this called Zeno's paradox.
E says) Hey D, did you make that up?

The answer depends on whether the product

converges or not
After much computation, I observed that it was converging to 0 very slowly.

After googling for such a product, I came across the beautiful Euler's product formula for Riemann zeta function-

Taking s=1, we are assured that the pie is completely eaten, but I wonder who gets the last piece!

bobbym · 2011-01-25 23:17:02

Hi gAr;

After A and B eat how much of the pie is left?

gAr · 2011-01-25 23:36:11

Hi,

After A and B eat, 1/3rd of pie is left, right?! Otherwise I might have misunderstood the problem.

bobbym · 2011-01-25 23:41:24

Yes, I think your use of the product of the primes is correct. That looks like you have the first convincing proof that the pie will be gone. Very good!

Math Is Fun Forum

#451 2011-01-25 17:45:05

Re: What do you think?

#452 2011-01-25 17:48:28

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#453 2011-01-25 17:48:39

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#454 2011-01-25 17:55:14

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#460 2011-01-25 18:27:14

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#461 2011-01-25 18:27:43

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#462 2011-01-25 18:29:10

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#464 2011-01-25 18:38:02

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#465 2011-01-25 18:40:26

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#466 2011-01-25 19:29:13

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#467 2011-01-25 19:40:35

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#468 2011-01-25 19:42:42

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#470 2011-01-25 19:58:33

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#472 2011-01-25 23:00:59

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#473 2011-01-25 23:17:02

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