Questions tagged [gambling]

For questions about the mathematics behind gambling, such as the expected value of a game or the efficiency of a gambling strategy.

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Random Walk with a reflecting barrier

I've got stuck in the below question as I do not know how to handle the reflecting barrier. We have a random walk with N+1 states from 0 to N. State N is an absorbing state and state 0 is a reflecting ...
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How to calculate the average number of turns until going bust for a simplified blackjack style game?

If for each turn, you draw a random number from a uniform distribution with a range between 1 and 20 inclusive and you add this number to your total sum, what is the average number of turns it would ...
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4 votes
2 answers
109 views

Best strategy to reach $500 for a gambling situation in a casino

Suppose a gambler has \$100 to start with. Each time he/she has 0.4 chances of winning and 0.6 chances of losing a bet. If he/she wins he gets twice the money he put in and loses what he bet if he ...
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Probability of Successive Lotto Games

My wife and I were both terrible at probability at school. So, we pretty much gave up straight away on this one. I wanted to know how the odds in a game of chance change (if at all) as more games are ...
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-1 votes
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What is market width in sport betting?

Examles: If the odds are -141/+123 ,then the market width is 141-123 = 18 cents If the odds are -110/-114 ,then the market width is 10+14 = 24 cents Why is that? How can Market width be used as an ...
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2 votes
1 answer
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Gambler’s Ruin Problem: how to calculate different probabilities?

I need to resolve a gambler's ruin problem so I have been looking into problem number 5 in the following pdf, which is similar to mine and it is resolved: https://www.webpages.uidaho.edu/~fuchang/...
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1 answer
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betting in fair game over infinite horizon, is it possible to win?

If a gambler were to play in a fair game, lets say he wins/loses 1 dollar with equal probability in each step. Let $X_i$ denote the amount of money he has after $i$ steps. And he plays until he either ...
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Computing the profit given the probability of winning in a sport bet

Suppose that you can correctly predict the outcome of a tennis match (i.e. the winner) with probability $x > 0.5$. If you gamble on $N$ matches on some betting website, where odds are assigned by ...
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1 vote
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An Extension of Gambler's Ruin

Suppose there is a gambler who has $n$ dollars gambles for infinitely many times. The payoff of the $t^{th}$ round is $X_t$, which is an integer beween $-1000$ and $1000$. We know that $\mathbb{E}[X_t|...
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1 vote
1 answer
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probability of rolling repeated numbers

if you roll an x-sided die, n-number of times, what are the chances of getting r-number of repeated values? if I roll a twenty-sided die, eighty times, what are the chances that I'll roll a six twice ...
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3 votes
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Dutch Book explained incorrectly in a book?

In chapter 4 of Darrell P. Rowbottom's Probability, the author explains the Dutch Book: Imagine we're going to have a bet together, you and I, about whether something will happen. It could be about ...
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Expected Roulette Profit after n rounds

Assume I have an infinite bankroll and bet on black with an American roulette wheel , while using the "Martingale" strategy (doubling your betsize every time you lose). How can you go about ...
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Let N_a denote the average duration of the game for player A starting with capital a

The Screenshot of the Question The objective here is to prove that for $$ p \neq q $$ $${N_a = \frac{b}{2p-1} - \frac{a+b}{2p-1}\frac{1-(\frac{p}{q})^b}{1-(\frac{p}{q})^{a+b}}}$$ and for $$ p = q = 1/...
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1 vote
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Does the Kelly Criterion assume that all future bets will stay the same?

In most explorations of the Kelly Criterion I’ve seen, we’re deciding the % of our bankroll to apply to a bet under the assumption we will be repeatedly faced with the same bet many times. I’m curious ...
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Modified Gambler's Ruins: going all in in each toss and allowing ties with probability $1-p-q$

Consider a modified gambler's ruin problem where in each coin toss the player either wins one more coin with probability $p$ or lose all his coins with probability $1-p$. a) Give an equation $P_m$ of ...
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4 votes
1 answer
69 views

Geometric growth rate and Kelly's Criterion question

In the Wikipedia page about Kelly Criterion, the author calculated the expected wealth after N bets as $$W * (1+g)^N$$ where $W$ is the initial wealth, and $g$ is the expected geometric growth rate. ...
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One period Binomial model - why is return function needed at "present time" for no arbitrage condition?

One period binomial model considers asset prices $S(i)$ where $i=0,1$ where $S(0) = S$ and we have either $S(1) = Su$ or $S(1) = Sd$, where $0 < d< u$, and nominal interest rate per period is $r$...
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Probability of reaching $\$0$ account balance before carrying out $N$ trades, risking $R_n$ each trade?

Suppose the trader starts with an account balance of $\$100$. For each trade $n=1,2,3,...$ (each trade is closed with its profit/loss reflected immediately in his account balance before starting the ...
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Are there any known martingale methods that can handle a limited budget?

Quick summary of martingale method: Bet X, if you lose next time bet 2 times X. You will cover all your losses on the next win. So the martingale method seems quite good on paper but doesn't work in ...
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2 votes
0 answers
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Indeterministic Rock, Paper, Scissors?

Suppose we play Rock, Paper, Scissors but the outcome isn't deterministic and guaranteed. If you play Rock and I play Scissors, you will only beat me with some probability in $[0,1]$. Likewise for the ...
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Maximizing Arbitrage Opportunity - Optimal Betting

There are two possible outcomes for a sporting event — Team A wins, or Team B wins. How can I calculate the maximum profit given the following arbitrage opportunity? Sportsbook 1 (Outcome Team A wins)...
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Maximize betting profit

A player makes a bet of $1 everyday and wins it to $2 or loses it to $0. The probability of ...
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1 answer
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Maximize the return of gambling with unknown distribution given samples

Given that I played a game with unknown probabilities and odds (actually they are some trades in reality) and I recorded the results into a spreadsheet. How can I calculate the best allocation in each ...
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-1 votes
2 answers
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Does the casino have an edge on a commission free 50/50 luck game?

I have an interesting discussion with someone on Youtube: Assume a commission free roulette (without the number 0) so you have a true 50/50 chance on black or red. Also assume that all players will ...
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2 votes
2 answers
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Betting system and upper bounds

Define $X_0=\alpha\in(0,1)$ the initial capital and $X_n$ as the remaining capital after each game. A player bets $1-X_n$ if $X_n>1/2$ and $X_n$ if $X_n\leq 1/2$ such that each game is a Bernoulli$(...
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0 votes
2 answers
140 views

Gambler's ruin problem but with many extra conditions

$A$ and $B$ play a series of games. They stop playing the game when the total number of wins of player $A$ is four greater than that of the player $B$ or when the total number of wins of player $B$ is ...
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51 views

Probability calculation in a gambling game

The problem I am trying to solve is the following: We are playing a game every day for 100 days. Assuming the winrate is always 75%. Every day you have the option to either: Play once with the ...
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2 votes
1 answer
119 views

Kelly betting: why do we maximize the expected value of the logarithm of wealth?

Introduction The Kelly betting criterion is a betting strategy for repeated games of chance which works by wagering a fixed proportion of one's bankroll each time. That is, suppose I play a game of ...
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Probability of M&Ms matching colors

In my office, we have a small toy M&M vending machine that dispenses about a handful of M&Ms at a time. My coworker began a game of asking me if there at least 2 M&Ms of the same color out ...
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Is it possible to calculate the odds for a one more goal to be scored in the first half from the odds of the teams winning the first half?

Let's say that at 35 minutes of the first half in one Bookmaker we find the following odds for the first half result: ...
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1 answer
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What is the probability of winning a bet on 30 of the 37 pockets at a roulette table?

At a roulette table, there is a wheel containing 37 possible pockets a ball can land on (each are marked 0, 1, 2, 3, etc. through to 36) and it is equally likely the ball could land on any of these ...
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0 votes
1 answer
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Calculating return to player in a slot machine, where a temporary bonus game raises the prizes

I am trying to program a machine that has 3 independent "reels", that are "rolled", and stops on 3 random icons The base machine is a basic slot with e.g. 100 possible combinations....
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1 vote
1 answer
87 views

Probability that the player wins a pass line bet with a 4 on the first roll

Precalculus textbook problem (self-study): In the game of craps, there are two ways a player can win a pass line bet. The player wins immediately if two dice are rolled and their sum is 7 or 11 . If ...
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1 vote
1 answer
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Game of draughts, expected value of first move advantage, part 2

This is a follow up to my question here: Game of draughts, expected value of first move advantage Here's a question from my probability textbook: $A$ and $B$ play at draughts and bet $\$1$ even on ...
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1 answer
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Game of draughts, expected value of first move advantage

Here's a question from my probability textbook: $A$ and $B$ play at draughts and bet $\$1$ even on every game, the odds are $\mu : 1$ in favor of the player who has the first move. If it be agreed ...
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1 vote
2 answers
121 views

Is the way to win at roulette to bet on green?

I apologize for the juvenile, and probably incorrect, math you are about to see. I have no intention of gambling. This post is made solely from curiosity. Anyways, I am underage, and thought of the ...
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1 vote
0 answers
24 views

Gambler fallacy and probability of tossing coins

Q. Three coins are tossed simultaneously 100 times with the following frequencies of different outcomes: If the three coins are simultaneously tossed again, compute the compute the probability of :  (...
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4 votes
1 answer
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Converting odds to probabilities

Gambling odds on sports betting are designated with a number in the set $(-\infty , -100) \cup [+100 , +\infty)$. If one places a wager of $w$ dollars at $p \in (-\infty , -100) \cup [+100 , +\infty)$ ...
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2 votes
2 answers
285 views

Expected Geometric Growth Rate (Kelly's Criterion)

The Wikipedia article for Kelly Criterion establishes its main formula using the expected geometric growth rate $r = (1 + fb)^p * (1 - fa)^q$, where $f$ is the fraction of an account (that starts with ...
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4 votes
1 answer
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Allocating money to different bets on loaded coins [closed]

I hope this question was not asked before. It is a very interesting problem I came across and for which I have not yet found a solution. If you know where to find it, please feel free to redirect me. ...
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Markov chain problem binomial distribution question (Gambler's ruin)

I have 16 euros in my pocket. I play heads or tails with $p=q=1/2$. I always bet on heads. If the coin turns heads i win 0 euros and if the coin flips tails i lose 1 euro. What is the probability of ...
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-1 votes
2 answers
206 views

payoff of a coin flip is $2^N$

We have a fair coin. The game ends when we get a head. the payoff is $2^N$ where N is the number of flips until we get the head. What is the expected value of the game? or, how much would you pay to ...
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How can I simulate a game of blackjack for different 'return to players'

I'm trying to simulate one game of blackjack, I'm doing it using code, but I think it's still a maths related question more than a programming one. My current working is this, for a return to player ...
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1 vote
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Markov chain with interpolation

I would like to solve this gambler problem using interpolation or any other technique that allows you to solve it with only pen and paper (without knowing the known gambler's problem formulas). Of ...
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0 votes
1 answer
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The probability of losing half by Kelly's criterion

According to "Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos And Wall Street", the probability of losing half initial money at some point using ...
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0 votes
1 answer
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Kelly Criterion with Taylor Approximation for Multiple Events

I am trying to understand how to approximate the kelly criterion when I have multiple winning events. The Taylor approximation says that $\log(1+fr) = fr + \frac{(fr)^2}{2}$ If I have 3 winning events,...
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0 votes
1 answer
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Betting Game on World Series

you and a friend are betting on individual games of the World series. For each game, if your team wins, you win a certain (positive) amount of money, and if your team loses, you lose that amount. You ...
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1 vote
0 answers
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Gambler's ruin with different strategies

At a certain casino game if you bet \$x you will loose your \$x with probability .505 (so your fortune will decrease by x) and win \$2x with probability .495 (so your fortune will increase by x). You ...
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Guessing Card Game

Consider a deck with 26 black and 26 red cards. You draw one card at a time and you can choose either guess on whether it is red beforehand or simply observe the result. If the card is red you get $1\$...
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3 votes
1 answer
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Probability of winning an arbitrarily long series of up/down wagers

Inspired by this question, I have been investigating the following problem: Suppose we start with $1$ unit of currency and place a series of wagers where with each wager we gain a unit of currency ...
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