# tricky algebra: With a fair coin, my opponent flips 50 times, with fair odds, ie,...

#### daveylibra

##### New member
Hi, I have searched and cannot find a similar question to this that has been answered...

Q. With a fair coin, my opponent flips 50 times, with fair odds, ie I double my bet on a win.
The average outcome is 25 wins, 25 losses.
However, if I make the assumption that I will win at least 15 times, how do I proportion my bets
to be ahead at the 15th win, whilst minimising bets?

Further, write an equation to illustrate this with e= expected wins, n= total flips.

I know e=15, n=50.
I think if c=expected wins at any point in the series, bet=1+(losses/c) should be right, but
it does not take into account n.

#### stapel

##### Super Moderator
Staff member
Q. With a fair coin, my opponent flips 50 times, with fair odds, ie I double my bet on a win.
Where does this "doubling my bet" thing come in? I see nothing in the preceding words which says anything about your bet, winning, rules, etc...?

The average outcome is 25 wins, 25 losses.
"The average outcome" being "of the coin tosses", rather than of "winning" or "bets"...?

However, if I make the assumption that I will win at least 15 times, how do I proportion my bets to be ahead at the 15th win, whilst minimising bets?
What other assumptions are you supposed to make? For instance, what happens when you "lose"?

When you reply, please include the full and exact text of the exercise, the complete instructions, and how you feel that this involves only "tricky algebra" rather than probability, etc. Thank you!

#### daveylibra

##### New member
Hi! Apparently...

1. for example, if I bet £1 and win, I get back £2. If I lose, I get back 0.

2. Lets assume I bet on heads every time. 25 heads, 25 tails = 25 wins, 25 losses (this is the average.)

3. I don't think there are any more assumptions. As before, when I lose I get back nothing.
I can bet any amount I choose on each flip.

Cheers!

#### daveylibra

##### New member
PS - This is the complete text of the exercise, and I wasn't sure whether to post in probability/statistics or algebra, or maybe calculus ??

#### daveylibra

##### New member

I think this problem could be unanswerable.
Unfortunately, I cannot PROVE its unanswerable....!

#### daveylibra

##### New member
Can anyone help with the above question?

I didn't think it was very complex... ? I think it must be answerable... ??

#### Dr.Peterson

##### Elite Member
Q. With a fair coin, my opponent flips 50 times, with fair odds, ie I double my bet on a win.
The average outcome is 25 wins, 25 losses.
However, if I make the assumption that I will win at least 15 times, how do I proportion my bets
to be ahead at the 15th win, whilst minimising bets?

Further, write an equation to illustrate this with e= expected wins, n= total flips.

I know e=15, n=50.
I think if c=expected wins at any point in the series, bet=1+(losses/c) should be right, but
it does not take into account n.

The problem is that your problem is not stated clearly enough to be able to answer it. That may be partly from lack of context.

Being stated in the first person, it does not look like a textbook problem; yet you eventually said, "This is the complete text of the exercise". If it is really an exercise from a book, you must state it exactly as given, and tell us what is being taught in that section, along with any assumptions you are told to make.

Context is everything when you ask a question.

#### daveylibra

##### New member
Its a question I am trying to answer for a friend...

#### tkhunny

##### Moderator
Staff member
There are only 32,768 cases for 15 tosses. Spreadsheets work.

1) Are you betting on each flip?
2) Do you reset back to the initial amount on a loss? Maybe just 1/2 and back to where you were on the previous bet.
3) What do you mean by "proportion".

The direct answer to you question, "How much more information could possibly be required?" is simply, "enough information for a well-defined problem". We're not there, yet.

Also, we're not really a consulting service. If you wan to LEARN, then we can help you AFTER you show your efforts.

Note: If there were an easy way to hedge a bet, why don't we have more rich gamblers?

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#### tkhunny

##### Moderator
Staff member
Note also, that you may be assuming one flip has ANYTHING to do with some other flip. If the flips are dependent, then you should bet LESS after each win.

#### daveylibra

##### New member
Hi 'tkhunny' - as to your questions

1) Yes, as we don't know whether any flip will be a win or lose.

2) No

3) "proportion" just means how much we bet each time.

All my efforts, I'm afraid, have not worked. Hence I tried this website.
I left school too long ago to remember my A level work.
Thought it would be a good question for people that like problem solving.

#### tkhunny

##### Moderator
Staff member
You earned my respect by simply answering my questions.

1) If you lose 15 in a row, you will lose only 15 - one each round.
2) If you win 15 in a row, you win 32,767 - twice as much for each of 15 rounds.
3) The unfortunate thing is the utter and complete loss of everything because of the bet-doubling. If you win 14 in a row and lose the last, you're still a dollar down after 15 rounds.
4) The probability of being ahead after 15 rounds is 29.66%. Mostly, this is a result of the odd wiping-out of everything after a single loss.
5) A cute histogram of outcomes

 Bin Frequency % -100 13,012 39.71% -50 4,752 14.50% -20 3,624 11.06% -10 992 3.03% 670 2.04% 10 454 1.39% 20 430 1.31% 50 1,080 3.30% 100 1,296 3.96% More 6,458 19.71%

"-50" means "a loss of 50 or more, but not so much as 100", eg. -50.00 - -99.99.
"100" means "a gain of 100 or less, but not so little as 50", eg +50.01 - 100.00

Like I said, spreadsheets work. It will take a substantially different betting strategy to improve the outcome.

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#### daveylibra

##### New member
Hi tkhunny

I wasn't thinking of using a spreadsheet, is this Excel?
I gather the formula used is, on a win, double the bet, on a lose, back to 1?

Cheers
Dave

#### tkhunny

##### Moderator
Staff member
Your instruction was to maintain the bet on a loss, not reset to 1.

I used MS Excel 2016. It was tedious, but not particularly difficult.

#### daveylibra

##### New member
I didn't give any instructions.

That was the question in the first place, how much to bet each time.

#### tkhunny

##### Moderator
Staff member
I didn't give any instructions.

That was the question in the first place, how much to bet each time.
Yes, you did.

Review:

2) Do you reset back to the initial amount on a loss? Maybe just 1/2 and back to where you were on the previous bet.

Hi 'tkhunny' - as to your questions

2) No