Mathematical Games and Gambling

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Mathematical Games and Gambling

• A project by Anthony Ward

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My Project

• My project has two parts
• How games can aid in the mathematical development
  process
• The origins of gambling- when and how to gamble.

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OMS Games

• Who Wants to be a Millionaire- using questions on
  current topics, helps teacher ascertain gaps in
  student knowledge
• Quick fire questions- develops quick thinking and
  mental arithmetic, and increase student
  participation in lessons.

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Card, Coin and Dice games

• In the Classroom- Develop ideas of probability
  and sample space.
• At A-Level and above- Develop ideas of
  probability distributions, expectation and
  combinatorics.

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THE PROBLEM

• Seven houses contain seven cats.
• Each cat kills seven mice.
• Each mouse had eaten
• seven ears of grain.
• Each ear of grain would have
• produced seven hekats of wheat.
• What is the total of all of these?

• Answer 742401

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Chevaliers Gambling Problems

• GAMBLE 1
• Attempt to roll a six in four tosses of a single
  die
• Slight advantage observed by Chevalier

• GAMBLE 2
• Attempt to throw two sixes in 24 tosses of 2
  dice.
• Slight disadvantage observed by Chevalier

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First Gamble- one 6 in four rolls

• If die lands on a 6 at least once in four rolls
  Chevalier wins
• If 6 is never thrown then he loses
• Each throw is independent so
• P(no 6 in 4 trials) (5/6)40.4823
• P(at least one 6)1-0.48230.5177

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Second Gamble- double six in 24

• If both die land on a 6 (a double six) in 24
  rolls the gambler wins
• If a double six does not occur gambler loses
• Each throw is independent so
• P(no double six)(35/36)240.5086
• P(at least 1 double six)1-0.5086)0.4914

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Expectations of the gambles

• Gamble 1
• E(X)1.(0.5177) (-1).(0.4823)0.0354

• Gamble 2
• E(X)1.(0.4914) (-1).(0.5086)-0.0172

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The Roulette Wheel

• On an American roulette wheel, there exists 38
  identical spaces in which the ball can land. The
  numbers 0, 00, and 1-36.
• Each pays out at odds of 1-35
• E(X)35.(1/38) 1.(37/38) -0.0526
• Conclusion Dont play!

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Do Gambles with positive expectations really
exist?

• Blackjack or 21 Card Counting
• Very complicated and beyond the scope of this
  project.

• Roulette strategies More investigation needed

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ANY QUESTIONS?