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Gambling Mathematics











































Probability and the Expected Return of Gambling

Understanding independent events (each bet is an independent event).

If a person wins two out of three 50 cent bets, how much money will that person win?

Comments for Gambling Mathematics

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Sep 17, 2013
Probability and Expected Value
by: Staff


Answer


Part I

Expected Value of a bet = the average winnings (or loss) per bet when you place a bet with the same odds over and over.


Expected Value = (Chance of Winning each bet) * (Amount you would win) – (Cost to place bet)


Expected Value of a Bet





The problem statement does not specify how much you will win for each bet that you do win.

Certainly, the amount you must win on a winning bet must be greater than 50 cents or there is no point in betting.


Expected Value = (2/3) * (Amount you would receive for each winning bet) - (.50)

Note: You must always invest $.50 for each bet (100 % of the time) because that is what it costs to place the bet.



Expected Value of a Bet when the odds of winning are 2 to 3




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Sep 17, 2013
Probability and Expected Value
by: Staff


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Part II


Since you will only win 2/3 of the time, the question is: On average, how much must each winning bet return to you to compensate for the $.50 it costs to place the bet?


Let x = Amount you receive for each winning bet

The break-even value of the winning amount is that amount which will cause the Expected Value to be $.00.

This amount bet can be calculated as follows:

Expected Value = (2/3) * (Amount you would receive for each winning bet) - (.50)


0 = (2/3) * (x) - (.50)

3 * 0 = 3 * ((2/3) * (x) - (.50))

0 = (3*2/3) * (x) - (3) * (.50)

0 = 2 * x - 3 * (.50)

0 = 2 * x - 1.50

0 + 1.50 = 2 * x - 1.50 + 1.50

1.50 = 2 * x + 0

1.50 = 2 * x

1.50 / 2 = 2 * x / 2

.75 = x

x = $0.75, the break even winning amount



Equation to determine how much money a winning bet must return to you in order to break even





Solve equation for break-even value






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Sep 17, 2013
Probability and Expected Value
by: Staff


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Part III



Solve equation for break-even value - finalize solution





the final answer is:

Given the odds listed in the problem statement, the amount returned for each winning bet must be greater than 75¢ to make the betting worthwhile.



Winning Bet must return more than 75 cents





The following illustration will demonstrate the dynamics of the process.

If you make three bets, you will invest $1.50 in the bets. If you win two bets, you must receive 75¢ for each win to break even.


1st bet: -$.50 + $.75 (cost 50¢, win 75¢ )

2nd bet: -$.50 + $.75 (cost 50¢, win 75¢ )

3rd bet: -$.50 + $.00 (cost 50¢, win 0¢ )
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Expected Value = -$1.50 + $1.50 = 0 (cost $1.50, win $1.50)
(you just break even)






Thanks for writing.

Staff
www.solving-math-problems.com


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