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# Although everyone who gambles at all probably tries to make a quick mental marginal analysis of the game, in depth analysis of the figures shall reveal how a rational player reacts to better odds, or a lower entry price, or a higher potential payout.

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Introduction

## Introduction

Although everyone who gambles at all probably tries to make a quick mental marginal analysis of the game, in depth analysis of the figures shall reveal how a rational player reacts to better odds, or a lower entry price, or a higher potential payout.  I also think it’s important to know at least a little bit about the gambling industry, seeing as it nets \$6.3 billion a year in revenue in Canada while net annual revenue from alcohol and tobacco sales in Canada is \$5.9 billion .  When searching for the relationship between Entry Price (E) into a game, the odds of winning that game (O), and the payout of the game (P), one must look at different games that incorporate these three variables.  There are three cases to be examined: Case 1 can be represented by a lottery, where E is set, and both O and P are known and the player tries to match numbers.  Case 2 can be represented by a simple game of chance, which is used in The St. Petersberg Paradox, where E is constant, but O is not set, therefore P is infinite.  Case 3 can be represented by a draw, where E is set, and both O and P are known.  A process of linear regression will be used to determine the actual relationship between P and O for each game.

Analysis

Case 1

The lottery chosen is from Ontario, lotto 649.

Middle

Table 5: Data for Case 3

 Prize Value (\$) Number of Prizes Odds of Winning 10 000 5 1/44 400 5 000 10 1/22 200 1 000 25 1/8 880 500 125 1/1 776 250 500 1/444 200 3015 1/73.631840796 150 5365 1/41.3793103448

But all of the prizes shall be included when calculating the expected value (except the values of all of the cars shall be added together for presentational purposes) thus expected value = + + + + = -\$66.6499414414

The odds of winning anything are Payoff is inversely proportional to the Odds of winning, as shown below:  Fig. 3:

Once again, a process of regression will be used to find the function of the relationship between Payoff P and Odds of winning (O).  Because the data points look to the naked eye as if there could be a linear relationship, my first estimate will be linear in nature, or be in the form: P = Table 6: First Estimate for Case 3

 Reciprocal of Odds Actual Payout Estimated Payout Difference Squared 44 400 \$10 000 \$9861.76 19110.30 22 200 \$5 000 \$4913.38 7503.02 8 880 \$1 000 \$1944.35 891796.92 1 776 \$500 \$360.87 19357.16 444 \$250 \$63.97 34607.16 73.631840796 \$200 -\$18.59 47781.59 41.3793103448 \$150 -\$25.78 30898.61 Total = 1051054.76

But since it does appear to have a slight curve, perhaps a quadratic equation would be more accurate:

P = + Table 7: Second Estimate for Case 3

 Reciprocal of Odds Actual Payout Estimated Payout Difference Squared 44 400 \$10 000 \$9925.92 5487.85 22 200 \$5 000 \$4509.12 240963.17 8 880 \$1 000 \$1732.17 536072.91 1 776 \$500 \$396.22 10770.29 444 \$250 \$156.96 8656.44 73.631840796 \$200 \$91.07 11865.74 41.3793103448 \$150 \$85.34 4180.92 Total = 817997.32

This is an improvement on the linear equation, and it is a good representation of the data.

Comparison

A major difference between these three games is the entry price.  In the lotto 649 lottery, it is a relatively small \$1, with a possibility of winning \$2 000 000.  The Heart and Stroke Lottery has a somewhat higher entry price of \$100, and the most one stands to win is \$1 000 000.

Conclusion

P = + and P = + When these equations are set as equal a quadratic in the reciprocal of the odds is formed:

9.99999981  = 0

This quadratic however has no positive roots; this is because Case 3 is always a better game to play when only considering the odds.  But it must be recalled that the function found to represent Case 3 was based on smaller numbers, thus it is misleading to compare it to other cases.

These three games do encompass most of the variables involved in gambling.  They have been examined and analyzed thoroughly to show relations between the three major variables, P, E, and O.  I have made some conclusions based upon my observations; firstly , secondly that E.  Thus every game of chance which involves these three variables revolve around the same proportionality: .  Therefore, the payoff of a game must be equal to a constant c multiplied by the entry price divided by the odds, or .  I theorize that the magnitude of this constant c determines the attractiveness of games of chance.  After finding the odds of the lotteries, I do wonder why people spend their money on them; it is simply a voluntary taxation in my opinion.  Another question that arises is, if someone were to run a business based on the game used in Case 2, would it be profitable?  And at what price would people be willing to enter, even though mathematically it has no bearing on their long term success in the game?

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

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