CASINO'S GAME

     

     Let us find out how the casino survives with mathematics.
     Say, your friend invite you for a game of dice.  You must bet (wager) 2 dollars.  If you roll 'six' you will get back 8 dollars.  The game will go on for 30 rounds.  All sounds good.
     The probability of rolling 'six' is 1/6.  Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5.  That is, you are expected to win 5 rounds out of 30.  Hence your gain will be 5 * 8 =40 dollars.  ok.  This also implies that you will loose 25 rounds.  Hence your loss will be 25*2 =50 dollars.  Your net gain will be
gain-less = 40-50 = -10 dollars.
For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars.  There will be a loss of -1/3 or 0.33 dollars per round.  It is not a fair game.
    Let us make a simple formula to calculate  'Pay out per round\.
The probability for a win = p
The pay-out in case of win = V
No. of rounds = n
The expected no.of wins = p.n
The total pay out =p.n.v
 The wager or your bet =w
Expected no.of losses =(1-p)n
Expected loss of money =(1-p)n.w
 Payout or gain or gain per round = pnv/n=pv
Loss per round =(1-p)nw/n = (1-p)w
Hence net pay out:
Gain-loss= pv-(1-p)w =N

This is the theoretical expected payout.  If only a few round are played, it will not be valid.  But a computer can simulate thousands of rounds.  And the computer calculated net pay out matches exactly with theoretical net pay out in the long run.  This is due to "law of large numbers"
    The casinos conducts thousands of rounds of a game in a day.  Hence they can easily calculate their gain and run their business.

  What is a fair game?  Let me offer you a dice game.  If you roll 'two six' in a row, you get 175 dollars.  The wager is 5 dollars.  A good deal. ok. The probability of rolling two six one after the other is p=1/36. Substitute in our formula and find out net pay out per round.
Net pay out =pv-(1-p)w
                      = 1/36*175-(1-1/36)5
                      =0
    Hence, it is fair game.  No gain or loss on the long run.  But , if  a few games are played, one will win and other will lose  
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