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The Magic Cafe Forum Index » » Puzzle me this... » » Comparing winning 1/2 to losing 1? (0 Likes) Printer Friendly Version

magicjohn2278
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Here's a concept that I posted on another thread to illustrate the relative effects of halving or doubling, I obviously didn't explain it very well, no-one took it up, or understood the point I was trying to make.

I'm going to offer you a game that you are unlikely to lose as everything is in your favour!

THE RULES
A game comprises a series of coin tosses.
The number of tosses in the series is always even.
The coin will fall heads or tails with equal probability.
That is to say, if there are 50 tosses in the series the coin will fall heads 25 times and tails 25 times, but not necessarily in HTHTHT order.
Every time the coin falls heads, I win
Every time the coin falls tails, I lose.

THINGS IN MY FAVOUR
I have unlimited resources and can never find myself in a situation where I can not pay what I owe you. - This works is in your favour too.

THINGS IN YOUR FAVOUR
You can choose the number of tosses in the series.
You start with any sum of money of your choosing.
You get to toss the coin. And determine the outcome.
…and this is the point…
Every time I lose, I double what you have.
Every time I win, you give me only half of what you have.

The game might start like this.
You start with 100
The coin falls heads, I win half what you have 50, you now have 50
The coin falls tails, I lose and give you a sum equal to what you have (50)
So you now have 100
I lose again, I give you another 100 - you have 200
I lose again and give you 200 - you have 400
I lose again and give you 400 - you have 800
And so on……..

So if the sum you hold is X, every time I lose, I give you X and you double your money. Every time I win you only give me half of X.

Obviously I am bound to lose in the long run but what is the most I can expect to lose? - and more to the point, why? Remember, you get to choose how the coin falls.
TomasB
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"That is to say, if there are 50 tosses in the series the coin will fall heads 25 times and tails 25 times, but not necessarily in HTHTHT order."

That is one strange coin. Maybe you mean "...about 25 times and tails about 25 times..."? The ordinary coin in my pocket actually has a probability above zero of falling tails up 50 times in a row when flipped absolutely randomly.

There is no limit on how much you can lose. But maybe you ask about your expected loss?

/Tomas
magicjohn2278
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Quote:
On 2006-03-31 05:59, TomasB wrote:
"The ordinary coin in my pocket actually has a probability above zero of falling tails up 50 times in a row when flipped absolutely randomly.

/Tomas


You aren't using an ordinary coin, you're making up the results to suit your most favourable outcome! (And besides I don't view SEK's as being "ordinary coins"!) Smile
magicjohn2278
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Quote:
On 2006-03-31 05:59, TomasB wrote:

There is no limit on how much you can lose. But maybe you ask about your expected loss?

/Tomas


So, assuming you played the game according to the rules, how much do I owe you?
How much did you start with, and how many times did you toss the coin?
TomasB
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Ok, the most favourable outcome would be to use an ordinary coin. A long line of people have lined up and they only want to play your game. Each one want the game to start at 100 dollars and each one only want to flip the coin twice.

The first player has the possible outcomes and the number after it is how much you loose in each game:

HH -75 dollars (you actually earn 75 dollars)
HT 0
TH 0
TT 300 dollars

Your expected loss with each player is 0.25*(-75)+0.25*0+0.25*0+0.25*300=56.25

So I do not think you would want to play the game with anyone tossing a real coin.

/Tomas
magicjohn2278
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The game played with the first and last players wasn't played according to the rules... The "game" with each Player comprised a "series" of 2 tosses...

RULE... "if there are 50 tosses in the series the coin will fall heads 25 times and tails 25 times,.... "
Andrei
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John - a coin tossed 50 times does NOT have a 100% chance of falling 25 heads 25 tails. Don't ignore the chance of it falling 24-26 and so on and so forth.

Andrei
magicjohn2278
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For the purposes of this game, we aren't using a real coin, the person I am playing is making up the results to suit himself! (Within the confines of the rules!)
TomasB
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Ok, I do not understand what the point of the game is. If you always walk forward as many steps as you move backwards you are back where you started, so you needn't have walked in the first place, or in your case, even suggest playing a game. Why would anyone take the time to play it? During that time you could work and earn money or go and see a short movie.

/Tomas
magicjohn2278
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As I see it, probably my least favourable scenario goes...

We start the game...
I lose lots of tosses in a row....
Opponent tosses coin....
Coin rolls down grate....
No coin so game is abandoned...
leaving me considerably out of pocket!

:spinningcoin:
magicjohn2278
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The point of the game is to illustrate that if you are offered a bet where you will either lose 2 or win 1, it isn't always a bad bet! (Pointless, but not bad!) - It depends on how it is phrased.
Your "walking" scenario illustrates this well, if I said to you that I was going to take X steps backwards, then 1/2X steps forwards, you wouldn't expect me to end up in the same place, but my game seems to illustrate that you do. (This is obviously because the value of X keeps changing.)

Remember your immediate reaction to the game, you said "There is no limit on how much you can lose."
.....so now would you care to reconsider?
TomasB
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Yes, if you force the same number of wins as losses you of course end up where you started with no loss and no wins.

/Tomas
magicjohn2278
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... even though you are only winning half of what you are losing!.....
Daegs
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Yes, but you are BETTER MAKING THIS BET if it will go 26-24 than you would lose going 24-26

THAT is what you cant prove because you are wrong!!!!

For *every* coin flip that goes past 50%, you are better off as you are winning more than you are losing!!!

I have sims to back this up... starting with $100 you win on average anywhere from a couple hundred thousand to a couple billion. Very rarely on runs of 10k(10k games with 100 tosses per game) would you ever lose money, and even then the most you can lose is $100.

So let's see... if we start with $100, and out of the 50 runs I get 4 more wins than you, so I get $1600.

When we play right again, and YOU get 4 more wins than me out of 50, then I get $6.125

So in the first game I win $1500, and in the second I only lose about $94, even though its the SAME DIFFERENCE IN WINS!!!!

I'd take the winnings of $1500 over a possible loss of $94 any day of the week... and that is only 4 tosses...

You can make nearly infinite with this game, you can only lose however much you start with... I'd take this game any day of the week.

It's NOT even.
magicjohn2278
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Quote:
On 2006-03-31 11:12, Daegs wrote:

It's NOT even.


You are misinterpreting the rules, the situations you describe are two separate "games". I was expecting to win and lose the same number of tosses in each game...
Steve Martin
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Seems you are trying to show that winning "1/2" is the same as losing "1" and that you think this is surprising.

It's not surprising at all, in the way that you are describing it.

In the game you describe (and taking a series of 2 coin tosses): when you win HALF of what I have (i.e. the "1/2" you are talking about), the FULL AMOUNT (i.e. the "1" you are talking about) that you will then lose is, of course, half of what I originally had.

In other words, you win a certain amount, and then you lose exactly that same amount.

So (obviously) in a series of any even length (equal wins and losses, as you seem to be specifying must be the case) both players will end up with what they started with.

I think you are confusing yourself by talking about "winning half" and "losing one" and believing there is something interesting in that.

What is the point of your question?
Any man who reads too much and uses his own brain too little falls into lazy habits of thinking.
Albert Einstein
magicjohn2278
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Sorry Steve, I was trying to make a point that I originally raised in a different thread, There I was trying to argue that winning double or losing half had an equal and opposite effect, obviously this was a futile argument and totally untrue, as others were quick to point out!

Since starting this thread, I've discussed the overall concept of the coin tossing game (at length) by p.m. and been advised that my scenario bears no relationship to the original thread.

Apologies for wasting your time.

(I still think the coin tossing thing interseting though!) Smile
Steve Martin
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NP
Any man who reads too much and uses his own brain too little falls into lazy habits of thinking.
Albert Einstein
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