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±1-random walk from 5 until 20 or broke

You can use symmetry here - Starting at $5$, it is equally likely to get to $0$ first or to $10$ first. Now, if you get to $10$ first, then it is equally likely to get to $0$ first or to $20$ first. ...
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±1-random walk from 5 until 20 or broke

It is a fair game, so your expected value at the end has to be $5$ like you started. You must have $\frac 34$ chance to go broke and $\frac 14$ chance to end with $20$.
• 377k

When to stop rolling a die in a game where 6 loses everything

Before deciding whether to stop or roll, suppose you have a non-negative integer number of points $n$. How many more rolls should you make to maximise the expected gain over stopping (zero)? ...
• 104k
Accepted

When to stop rolling a die in a game where 6 loses everything

In the last round you can get $\frac{1+2+3+4+5}{6}$ or lose $p\frac 1 6$, whenever the second is more than the first you should stop. So once you have scored more than 15 you should stop. If you score ...
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• 13.2k
Accepted

3 person bet based on the perceived likelihoods of an outcome

This is an interesting problem. I thought about it for some time and I believe I have a complete solution. First, let's see how we would answer the question if we had only 2 people. Say, Alice ...
• 3,175

±1-random walk from 5 until 20 or broke

All answers so far are great but some readers seem to feel they lack an intuitive explanation - and perhaps the maths to back it up. Consider that you have an equal chance of moving up or down along ...
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Kelly Criterion for simultaneous independent bets

Any elaborate answer appears to be helplessly convoluted. Nonetheless, Withrow (2007) concluded: When the number of bets is small, the optimal sizes of bet seem to be almost exactly proportional to ...
• 237

When to stop rolling a die in a game where 6 loses everything

There is unfortunately many mistakes in the calculation of the expectation proposed: 1) The expectation does not take into account the optionality of the player deciding to play or not. 2) Also ...
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Odds of 8 to 1 mean you make a bet of \$1, then you win \$8. You also receive your original \$1 back, so if you win an 8 to 1 bet, you pay \$1 and receive back \$9. In regards to your problem, the ... • 14.6k 6 votes Accepted When should I stop playing this dice game? Your calculation is incorrect because$2^{n}\cdot \left(\frac 56 \right)^n$is the expected profit of rolling$n$times and quitting. You should not subtract the$\frac 162^{n}$because the loss of ... • 377k 6 votes Best strategy to reach$500 for a gambling situation in a casino

Your result of $0.4^3=0.064$ is correct for strategy 1. As expected, it turns out to be better than strategies 2 or 3 but not sensibly adapted versions of any of the three strategies. Your ...
• 158k

Gambler's fallacy and the Law of large numbers

Any sequence has the same probability as any other, but there are more sequences that are "balanced" than any other given proportion. For example, if I flip a coin 4 times then there are 6 ways to get ...
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When to stop rolling a die in a game where 6 loses everything

It's not hard to see that the optimal strategy in this game is "stop on $k$ or higher," for some positive integer $k$. This is because what is optimal is only determined by your current score, not ...
• 77.5k
Accepted

Why don't billionaires (or multi-millionaires for that matter) use the Martingale betting system?

Two main reasons: It is actually unprofitable in the very long run, no matter how large one's finite wealth is. In your example, the player is one loss away from going bust as early as round 17 (edit:...
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The perfect gambler - would it work?

This is a gambling system called the "Martingale System" and it has a reasonable chance of working in the short term, but the issue is that in the long term you will always lose. if you just bet one ...
• 1,111
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The perfect gambler - would it work?

The flaw is that if you lose seven in a row you don't have enough coins to double your bet again. It is a losing bet, so the more you play the more certain you are to lose. No betting strategy can ...
• 377k
Accepted

How Much is "Large" in the Law of Large Numbers?

The Law of Large Numbers is not an empirical result. You cannot demonstrate this with your own finite-trial experiment. It is a mathematical statement which states that - if you're familiar with ...
• 19.6k
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Infinite Coin toss: Borel-Cantelli and Kolmogorov

It suffices to show that, with probability 1, B gives 1 dollar to A $n$ times in a row (or less if the game will end). We can split up $A_1,A_2,\dots$ into blocks of $n$, i.e., let $E_1$ be the event ...
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How many tickets should Paul buy?

I've found a proof for $n=\textbf{196}$. In fact, Paul can guarantee a third with the following strategy. Observe that if you consider the set $G=\{1, 2, ..., 49 \}$ as the union of three sets $A, B$ ...
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Accepted

A winning wager that loses over time

Here are all the possible outcomes if you stop after ten tosses of the coin. I have rounded the numbers for display, but the calculations were at the full precision of the software on which they were ...
• 99.5k
I would argue that the optimal initial betting is $\frac{2 - \sqrt{2}}{2} \sim 0.29$. To do so, we need a few arguments: Whatever the initial bet is that you use to go from $0$ to $+1$, the second ...