Wikipedia:Reference desk archive/Mathematics/2006 July 18

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[edit] Bluetooth Remote Control

Is there a way to use a PDA with bluetooth to control things like volume on a bluetooth-equipped laptop? Basically, could you use the bluetooth to make a PDA work like a remote control for, say, iTunes? Many thanks, --86.139.122.229 16:04, 18 July 2006 (UTC)

Perhaps a Google search would be helpful? --LarryMac 16:28, 18 July 2006 (UTC)

[edit] Probability

You play a game in which you have 99% chance of winning each time you play it. You have already played it 10,000 times and won every time. If you play the game again and win, then you will gain a trillion dollars. If you lose, then you will go into debt a billion dollars. The question is, should you keep playing, because the outcome of previous games do not affect the outcome of the next game, so it is likely to win? Or should you stop playing, because there is little chance of winning 10,001 times in a row, even if the chance of winning each game is 99%? --Yanwen 19:35, 18 July 2006 (UTC)

We don't know enough to answer the question. The first rule of gambling is "never risk more than you can afford to lose". Insurance evolved as a means to distribute risk, especially of shipping, so that no single party could lose too much. This would be an excellent gamble for a group, but perhaps more risk than most individuals could carry. --KSmrqT 06:50, 19 July 2006 (UTC)
First, it can be inferred from your description that the games are independent. This means that the results of the last 10,000 times have no effect whatsoever on the result of the next game. So the chance of winning the next game is 99% - you can be quite confident in your victory.
However, the decision of whether to play again in such circumstances is a different question entirely. Unless you are already a multi-bliionaire, a debt of 1 billion dollars will pretty much end your life. On the other hand, earning a trillion dollars is not that different from earning a billion - both are much more than you will ever need in your entire lifetime. Because of the 1% chance of losing, and the devestating consequences, most people would conside this deal a bad one. -- Meni Rosenfeld (talk) 19:47, 18 July 2006 (UTC)
I disagree that being a billion dollars in debt would "pretty much end your life". Just declare bankruptcy and get on with life, it's only a bit of an inconvenience. StuRat 22:39, 20 July 2006 (UTC)
Whatever evidence convinced me that I have a 99% chance of winning in the first place, I would take that evidence to a large insurance company. (And first, I would hire an excellent lawyer team of lawyers.) After working out a deal to ensure that I earn money regardless of the outcome, then I would play the game. Melchoir 19:54, 18 July 2006 (UTC)

I agree that you everytime you play you have a 99% chance of winning: it does not matter how many times you play. To think otherwise would be the Gamblers fallacy. For example, if you keep flipping coins, even if you have got four heads in a row, you still have a 50% chance of getting a head next time.

I'm not sure how big a trillion is - I think its a lot more than a billion. In that case if you would only be at risk the first time you played the game, since your winnings from the first game would cover the debt if you lost. So I would keep playing until I had as much money as I pleased.

If however you mean you would have all your winnings wiped out and also be in debt if you lost, then I would decide what risk I could accept of losing and play that number of times. You chance of winning over all n plays is 0.99^n . So if I was happy to accept a 10% chance of loss, in other words a 90% chance of winning, then

0.90=0.99^n

therefore

n=log(0.90)/log(0.99)

Or at least I think it is.

You could read the article Kelly criterion about making decisions regarding how much to bet, although as you do not say how much is wagered and what your stake is, it does not apply to the game you have described. --62.253.48.198 21:39, 18 July 2006 (UTC)

Yanwen, in short, you are likely to win (this is trivially true since you already stated that you had a 99% chance of winning). If we do assume "the outcome of previous games do not affect the outcome of the next game" like you stated, then the likelihood of you winning the 10,001 game is the exact same as every game you've played before (and will play in the future). So your worry that "Or should you stop playing, because there is little chance of winning 10,001 times in a row" is unfounded. If you wanted to further analyze the situation based on your utility theory (where you could figure out just how much you'd welcome the consequences of each outcome and such). But in short yes you should take the bet. If you want a formal decision theoretic answer I could dust off my decision theory book if you desired, just ask.--droptone 00:14, 19 July 2006 (UTC)

I might add that your expected winnings in this situation are $1 trillion ($1000000000000) times 99% minus $1 billion ($1000000000) times 1%, which is $989990000000. So if this scenario transpires many times, you will on average win $989990000000 per itineration.Amanaplanacanalpanama 00:31, 19 July 2006 (UTC)

The chances that you win 10000 times in a row is ridiculus. I think there is something wrong with the game itself.

>>> 0.99**2
0.98009999999999997

>>> 0.99**10000
2.2487748498162805e-44

Ohanian 00:46, 19 July 2006 (UTC)

Ridiculous scenarios and extreme cases are very helpful in thinking about stuff like this. JackofOz 02:02, 19 July 2006 (UTC)

The answer to your question is that you already know you're being lied to. Somebody is telling you there is a 1% chance of losing each time you play. You play 10,000 times, expecting to lose 100 times. Instead, you lose 0 times. That means (with incredible statistical certainty) that you were being lied to about the probabilities. If I tell you there's a 90% chance of you exploding at any given minute, and you live 1,000,000 without exploding, I was lying. It's that simple. 82.131.187.228 18:54, 20 July 2006 (UTC).

Assuming, of course, that being lied to is a possibility in this problem. Nothing in the problem setup is actually contradictory, it's merely very, very improbable. Playing 10,000 games at 99% chance to win and winning every one could happen, it's just statistically (very) unlikely. Sort of like if someone said, "Suppose you toss a hundred coins, and they all land on-edge." Not impossible, just improbable. Maelin 14:09, 24 July 2006 (UTC)