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January 6

Homomorphisms between finitely generated modules

Let R be a commutative ring and M and N be 2 finitely generated R-modules. Is it possible for $Hom_{R}(M,N)$ to not be finitely generated? Is it Noetherian (hence finitely generated) if N is Noetherian? GeoffreyT2000 (talk, contribs) 17:54, 6 January 2017 (UTC)[reply]

Betting with the Martingale system, when events are not independent

You can choose between two types of games: one where the results are independent (like roulette), and, another where events are not independent from each other, and there are indeed winning and losing streaks (sports).

Does your choice matter if following a betting 'strategy' like Martingale's? — Preceding unsigned comment added by 123abcnewnoob (talk • contribs) 20:02, 6 January 2017 (UTC)[reply]

I've put in a link to the system. Loraof (talk) 21:40, 6 January 2017 (UTC)[reply]

I would say it makes no difference, if you follow the “system” like so:

The idea of the Martingale system is you bet e.g. $1 to win $1 (with odds of 50:50 such as flipping a fair coin). If you lose then you double bet to $2, so if you win $2 it wipes out your initial loss and you are $1 ahead. Keep doubling until you win.

You can use this with odds other than 50:50. Simply adjust your stake, so you win enough to wipe out your initial loss and come out $1 ahead. The odds do not need to be the same each time, you can adjust your stake each time. They can be not independent of each other, which will be reflected in the odds. It is the odds of each bet that matters, and each time you bet enough to wipe out your losses and win the initial $1.

The problem is the same as the classic example of a 50:50 bet. You are betting 1, 2, 3, ... times to win $1. Each of these bets costs more than the last. Eventually you have a streak of bad luck, can’t afford the next bet and lose everything.--JohnBlackburne^words_deeds 21:57, 6 January 2017 (UTC)[reply]

Or, depending on the game, at some point your bet may reach the table limit, so you can not make your bet as large as it needs to be to cover your loss. CodeTalker (talk) 22:16, 6 January 2017 (UTC)[reply]

And what about if the player adapted to wins and losses? That is, if the player halved his bet by each loss, and doubled his bet by each win? Or maybe the other way round? Would this work differently? 123abcnewnoob (talk) 23:45, 6 January 2017 (UTC)[reply]

If the probabilities are not independent then you could use your knowledge of recent wins and losses to get a better estimate of the actual probability than if you used just percent wins. But presumably the person you're placing the bet with has the same information and would adjust the odds accordingly. Mixing in Martingales doesn't change this. --RDBury (talk) 03:41, 7 January 2017 (UTC)[reply]

Kelly criterion may be of interest. —Tamfang (talk) 07:22, 7 January 2017 (UTC)[reply]