• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14
  15. 15
    15
  16. 16
    16

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.

Extracts from this document...

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 billionimage00.png a year in revenue in Canada while net annual revenue from alcohol and tobacco sales in Canada is $5.9 billionimage01.png.  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.

...read more.

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 = image41.png

+image42.png+

image43.png+image44.png+

image45.png = -$66.6499414414

The odds of winning anything are

image46.png

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: image47.png

P = image49.png

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 = image08.png+ image09.png

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.

...read more.

Conclusion

P = image08.png+ image09.png and P = image14.png+ image15.png

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

9.99999981image16.pngimage17.png= 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; firstlyimage18.png, secondly that image20.png E.  Thus every game of chance which involves these three variables revolve around the same proportionality:image21.png.  Therefore, the payoff of a game must be equal to a constant c multiplied by the entry price divided by the odds, orimage22.png.  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?

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Core & Pure Mathematics essays

  1. Marked by a teacher

    Estimate a consumption function for the UK economy explaining the economic theory and statistical ...

    3 star(s)

    The final part summary the whole project. Appendix Table 1 EQ( 1) Modelling C by OLS (using project12min.xls) The estimation sample is: 1948 to 2002 Coefficient Std.Error t-value t-prob Part.R^2 Constant 13702.8 3000. 4.57 0.000 0.2824 Y 0.920727 0.007770 119. 0.000 0.9962 sigma 8845.94 RSS 4.14728156e+009 R^2 0.99624 F(1,53)

  2. Investigate the number of winning lines in the game of connect 4.

    Substitute 'a' which is 11 back into (2) 17=11x4+b 17=44+b b=17- 44 b = 27 b=-27 substitute 'a' and 'b' back into original equation. C=11H-27 that is the equation for the number of connects in a Nx5 box. But since the first 2 heights didn't follow the pattern we didn't use them in the equation so this equation doesn't work for them.

  1. Estimate a consumption function for the UK economy explaining the economic theory and statistical ...

    can see from comparing figures 5 and 6 there is virtually no difference after adding this variable as the co-efficient of uncertainty this means that uncertainty is either being modelled incorrectly or the argument that it played a major in the change of consumption over the late 80's was incorrect.

  2. Functions. Mappings transform one set of numbers into another set of numbers. We could ...

    can order that is equal to or larger than the denominator ? The remainder is written as a fraction ==> There are two methods for doing this ? Polynomial long division ? Remainder theorem ==> Remainder has to have a lower power than the divisor Let F(x)

  1. Math Portfolio Type II - Applications of Sinusoidal Functions

    Calculate the vertical stretch factor that would be required in the transformation of the graph of function f into the graph of function g. The amplitude that represents the time of sunrise in Toronto is 1.627 and the amplitude that represents the time of sunrise in Miami is 0.846.

  2. Math assignment - Families of Functions.

    The y-intercept for an absolute value function is b, and it is not to be forgotten that b is inside the absolute value symbols. So, you calculate the graph through using the three points that you have got, the y-intercept, the vertex, and the point, symmetric to the y-intercept.

  1. Explore and review the basic physics of how different things relate to one another.

    Table 1 has the results with the number of loonies being substituted by their individual masses. Table 1. Extension of a rubber band with loonies Loonies added Mass (g) Total weight (N) Length (mm) Extension (mm) 0 0 0 56 0 1 6.97 0.0684 58 2 2 6.97 0.1368 58.5

  2. The Rational Zeros

    We obtain the smallest positive root as . These set of results corroborate the assertion on the previous section, while the second and fourth term are kept constant, the numerator of the smallest positive root remains the same, while the denominator assumes the value of the coefficient of the first term.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work