Fairness of U.S. Quarters, Nickels, and Pennies

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Fairness of U.S. Quarters, Nickels, and Pennies
By Nate Golomb
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Are U.S. Coins Fair?

• Coins are used to decide disagreements with a
  flip or spin, to determine the start of a
  sporting event, coin games involving
  spinning/flipping, etc.
• Are we sure all coins are fair and acceptable for
  these ordeals?
• To test the question, spinning a coin on a flat
  surface will be used because an unfair weighting
  has less an affect when flipped versus when the
  coin is grounded.

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Rumor has it

• Quarters are said to be fairly weighted the
  ratio of heads/tails should be 50.
• Thus, if fairly weighted, heads will come up the
  same number of times as tails.
• Nickels and Pennies are rumored to have more
  weight on the heads side causing tails to come
  up more often when spun on a flat surface.

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Experiment

• The experiment run used five quarters, nickels,
  and pennies of all different years to represent a
  simple random sample of all respective coins in
  circulation.
• Each coin was spun 30 times on a flat surface for
  a total of 150 possible outcomes.
• The number of head and tail outcomes were
  recorded and proportion significance tests were
  conducted against the expected ratio of 50.

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Spinning Coin Demonstration
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Quarter Results

• Proportions Heads 77/150 Tails 73/150
• Percentages Heads 51.33 Tails 48.67

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Nickel Results

• Proportions Heads 64/150 Tails 86/150
• Percentages Heads 42.67 Tails 57.33

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Penny Results

• Proportions Heads 51/150 Tails 99/150
• Percentages Heads 34 Tails 66

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Quarter Significance Test

• Null Hypothesis p 0.5 Alternative Hypothesis
  p lt 0.5
• Test Statistic z 0.33
• P-Value p 0.628
• Conclusion We cannot conclude that the quarter
  is unfairly weighted causing one side of the coin
  to come up more than the other when spinning it
  on a flat surface.

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Nickel Significance Test

• Null Hypothesis p 0.5 Alternative Hypothesis
  p lt 0.5
• Test Statistic z -1.80
• P-Value p 0.036
• Conclusion We can conclude that the nickel is
  unfairly weighted on the heads side causing tails
  to come up more often when spinning it on a flat
  surface.

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Penny Significance Test

• Null Hypothesis p 0.5 Alternative Hypothesis
  p lt 0.5
• Test Statistic z -3.92
• P-Value p 4.44298 x 10-5
• Conclusion We can conclude that the penny is
  unfairly weighted on the heads side causing tails
  to come up more often when spinning it on a flat
  surface.

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Overall Conclusions

• Both pennies and nickels are deemed to be
  unfairly weighted and unable to provide a fair
  50-50 trial or game.
• The heavier heads side of the coin causes tails
  to come up more often when spun on a flat
  surface.
• Dimes were not tested because they have not been
  rumored as unfair, however, they should be tested
  before assuming they are fair.
• Thus, Quarters will provide a tested, reliable,
  and fair 50-50 trial or game.