# 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

 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) 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. -------------------------------------------------

 Sep 17, 2013 Probability and Expected Value by: Staff ------------------------------------------------- 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 -------------------------------------------------

 Sep 17, 2013 Probability and Expected Value by: Staff ------------------------------------------------- Part III 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. 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¢ ) ----------------- 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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