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The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Whats in your wallet?! Lara Pudwell Valparaiso University Butler University Mathematics Colloquium October 11, 2019 Whats in your

  1. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun The simple spender Eric usually uses his debit card... except when he spends $5.20 cash on a latte. What does his wallet look like? Start: 0 cents Then: 100-20 = 80 cents Then: 80-20 = 60 cents Then: 60-20 = 40 cents Then: 40-20 = 20 cents What’s in your wallet?! Lara Pudwell

  2. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun The simple spender Eric usually uses his debit card... except when he spends $5.20 cash on a latte. What does his wallet look like? Start: 0 cents Then: 100-20 = 80 cents Then: 80-20 = 60 cents Then: 60-20 = 40 cents Then: 40-20 = 20 cents Then: 20-20= 0 cents ...and repeat! What’s in your wallet?! Lara Pudwell

  3. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun The Coin Keeper 1 The Simple Spender 2 The Gamblers 3 The Small Spender 4 The Whole Shebang 5 ...and More Fun 6 What’s in your wallet?! Lara Pudwell

  4. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun On the planet Markovia , coins aren’t for spending money. They change colors and they’re for playing the lottery. Original coin has a 50% chance of being red, 50% chance of being blue. For every round of the lottery, ◮ 1 / 3 of red coins turn blue. ◮ 3 / 4 of blue coins turn red. ◮ After 10 rounds, all the players with blue coins share the prize. Shorthand: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 What’s in your wallet?! Lara Pudwell

  5. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Shorthand: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Question: If I have a blue coin now, what’s the probability that it will be red in the next round, and blue in the round after that? What’s in your wallet?! Lara Pudwell

  6. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Shorthand: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Question: If I have a blue coin now, what’s the probability that it will be red in the next round, and blue in the round after that? Answer: 3 4 · 1 3 = 1 4 = 0 . 25 What’s in your wallet?! Lara Pudwell

  7. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Shorthand: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Question: What is the probability of starting with a blue coin and having it stay blue for all 10 rounds? What’s in your wallet?! Lara Pudwell

  8. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Shorthand: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Question: What is the probability of starting with a blue coin and having it stay blue for all 10 rounds? � 1 � 10 = Answer: 1 1 2 · 2097152 ≈ . 0000004768371582 4 What’s in your wallet?! Lara Pudwell

  9. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Shorthand: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Question: If I play the lottery, what’s the probability that I’ll have a blue coin at the end of 10 rounds? What’s in your wallet?! Lara Pudwell

  10. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Shorthand: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Question: If I play the lottery, what’s the probability that I’ll have a blue coin at the end of 10 rounds? Answer: Markov chains! What’s in your wallet?! Lara Pudwell

  11. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Side note: Matrix Multiplication We have: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Represent this with two matrices: � 1 Initial state matrix: v (0) = 1 � 2 2 � 2 / 3 � 1 / 3 Transition probability matrix: P = 3 / 4 1 / 4 What’s in your wallet?! Lara Pudwell

  12. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Side note: Matrix Multiplication We have: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Represent this with two matrices: � 1 Initial state matrix: v (0) = 1 � 2 2 � 2 / 3 � 1 / 3 Transition probability matrix: P = 3 / 4 1 / 4 Here’s how to multiply a 1 × 2 matrix times a 2 × 2 matrix: � C � D � � � � A B × = ? ? E F What’s in your wallet?! Lara Pudwell

  13. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Side note: Matrix Multiplication We have: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Represent this with two matrices: � 1 Initial state matrix: v (0) = 1 � 2 2 � 2 / 3 � 1 / 3 Transition probability matrix: P = 3 / 4 1 / 4 Here’s how to multiply a 1 × 2 matrix times a 2 × 2 matrix: � C � D � � � � A B × = AC + BE ? E F What’s in your wallet?! Lara Pudwell

  14. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Side note: Matrix Multiplication We have: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Represent this with two matrices: � 1 Initial state matrix: v (0) = 1 � 2 2 � 2 / 3 � 1 / 3 Transition probability matrix: P = 3 / 4 1 / 4 Here’s how to multiply a 1 × 2 matrix times a 2 × 2 matrix: � C � D � � � � A B × = AC + BE ? E F What’s in your wallet?! Lara Pudwell

  15. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Side note: Matrix Multiplication We have: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Represent this with two matrices: � 1 Initial state matrix: v (0) = 1 � 2 2 � 2 / 3 � 1 / 3 Transition probability matrix: P = 3 / 4 1 / 4 Here’s how to multiply a 1 × 2 matrix times a 2 × 2 matrix: � C � D � � � � A B × = AC + BE AD + BF E F What’s in your wallet?! Lara Pudwell

  16. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Side note: Matrix Multiplication We have: red blue 1 1 start 2 2 2 1 red 3 3 3 1 blue 4 4 Represent this with two matrices: � 1 Initial state matrix: v (0) = 1 � 2 2 � 2 / 3 � 1 / 3 Transition probability matrix: P = 3 / 4 1 / 4 Here’s how to multiply a 1 × 2 matrix times a 2 × 2 matrix: � C � D � � � � A B × = AC + BE AD + BF E F What’s in your wallet?! Lara Pudwell

  17. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun � 1 Initial state matrix: v (0) = 1 � 2 2 � 2 / 3 � 1 / 3 Transition probability matrix: P = 3 / 4 1 / 4 v ( i ) = probability matrix after i steps = v (0) P i . What’s in your wallet?! Lara Pudwell

  18. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun � 1 Initial state matrix: v (0) = 1 � 2 2 � 2 / 3 � 1 / 3 Transition probability matrix: P = 3 / 4 1 / 4 v ( i ) = probability matrix after i steps = v (0) P i . � � 1 � 1 v (1) = 2 · 2 3 + 1 2 · 3 2 · 1 3 + 1 2 · 1 � � � 4 4 � 17 7 � � � = ≈ 0 . 7083 0 . 2917 24 24 What’s in your wallet?! Lara Pudwell

  19. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun � 1 Initial state matrix: v (0) = 1 � 2 2 � 2 / 3 � 1 / 3 Transition probability matrix: P = 3 / 4 1 / 4 v ( i ) = probability matrix after i steps = v (0) P i . � � 1 � 1 v (1) = 2 · 2 3 + 1 2 · 3 2 · 1 3 + 1 2 · 1 � � � 4 4 � 17 7 � � � = ≈ 0 . 7083 0 . 2917 24 24 v (2) = v (0) P 2 = v (1) P ≈ � � 0 . 6910 0 . 3090 What’s in your wallet?! Lara Pudwell

  20. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun � 1 Initial state matrix: v (0) = 1 � 2 2 � 2 / 3 � 1 / 3 Transition probability matrix: P = 3 / 4 1 / 4 v ( i ) = probability matrix after i steps = v (0) P i . � � 1 � 1 v (1) = 2 · 2 3 + 1 2 · 3 2 · 1 3 + 1 2 · 1 � � � 4 4 � 17 7 � � � = ≈ 0 . 7083 0 . 2917 24 24 v (2) = v (0) P 2 = v (1) P ≈ � � 0 . 6910 0 . 3090 and v (10) = v (0) P 10 ≈ � � 0 . 6923 0 . 3077 After 10 rounds, you have a 30.77% chance of winning the Markovian lottery! What’s in your wallet?! Lara Pudwell

  21. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chain behaviors: 1 absorbing – there are states where you can get stuck for forever. 2 cyclic – there exist some states where you cycle between them for forever. 3 regular – for some positive integer n , P n has no zero entries. What’s in your wallet?! Lara Pudwell

  22. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chain behaviors: 1 absorbing – there are states where you can get stuck for forever. 2 cyclic – there exist some states where you cycle between them for forever. 3 regular – for some positive integer n , P n has no zero entries. The Markovian lottery is regular . Question: What if we played the Markovian lottery for infinitely many rounds? What’s in your wallet?! Lara Pudwell

  23. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun For regular Markov chains Have transition probability matrix P . Want long term probability matrix L of ending up in each state. Big idea: LP = L (and the entries in L sum to 1.) What’s in your wallet?! Lara Pudwell

  24. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun For regular Markov chains Have transition probability matrix P . Want long term probability matrix L of ending up in each state. Big idea: LP = L (and the entries in L sum to 1.) � � 2 / 3 � 1 / 3 � � � Here: p r p b = p r p b 3 / 4 1 / 4 What’s in your wallet?! Lara Pudwell

  25. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun For regular Markov chains Have transition probability matrix P . Want long term probability matrix L of ending up in each state. Big idea: LP = L (and the entries in L sum to 1.) � � 2 / 3 � 1 / 3 � � � Here: p r p b = p r p b 3 / 4 1 / 4 Solve: 2 / 3 p r + 3 / 4 p b = p r 1 / 3 p r + 1 / 4 p b = p b p r + p b = 1 p r = 9 13 ≈ 0 . 6923 , p b = 4 13 ≈ 0 . 3077 What’s in your wallet?! Lara Pudwell

  26. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun The Coin Keeper 1 The Simple Spender 2 The Gamblers 3 The Small Spender 4 The Whole Shebang 5 ...and More Fun 6 What’s in your wallet?! Lara Pudwell

  27. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chains for coins In the land of simplicity there are 25-cent and 50-cent coins. All prices end in 0, 25, 50, or 75 cents. Possible wallet states? charged 0 25 50 75 start empty What’s in your wallet?! Lara Pudwell

  28. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chains for coins In the land of simplicity there are 25-cent and 50-cent coins. All prices end in 0, 25, 50, or 75 cents. Possible wallet states? charged 0 25 50 75 start empty empty { 25,50 } { 50 } { 25 } What’s in your wallet?! Lara Pudwell

  29. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chains for coins In the land of simplicity there are 25-cent and 50-cent coins. All prices end in 0, 25, 50, or 75 cents. Possible wallet states? charged 0 25 50 75 start empty empty { 25,50 } { 50 } { 25 } { 25 } { 50 } { 25,50 } What’s in your wallet?! Lara Pudwell

  30. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chains for coins In the land of simplicity there are 25-cent and 50-cent coins. All prices end in 0, 25, 50, or 75 cents. Possible wallet states? charged 0 25 50 75 start empty empty { 25,50 } { 50 } { 25 } { 25 } { 25 } empty { 25,50 } { 25,25 } { 50 } { 50 } { 25 } empty { 25,50 } { 25,50 } { 25,50 } { 50 } { 25 } empty What’s in your wallet?! Lara Pudwell

  31. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chains for coins In the land of simplicity there are 25-cent and 50-cent coins. All prices end in 0, 25, 50, or 75 cents. Possible wallet states? charged 0 25 50 75 start empty empty { 25,50 } { 50 } { 25 } { 25 } { 25 } empty { 25,50 } { 25,25 } { 50 } { 50 } { 25 } empty { 25,50 } { 25,50 } { 25,50 } { 50 } { 25 } empty What’s in your wallet?! Lara Pudwell

  32. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chains for coins In the land of simplicity there are 25-cent and 50-cent coins. All prices end in 0, 25, 50, or 75 cents. Possible wallet states? charged 0 25 50 75 start empty empty { 25,50 } { 50 } { 25 } { 25 } { 25 } empty { 25,50 } { 25,25 } { 50 } { 50 } { 25 } empty { 25,50 } { 25,25 } { 25,50 } { 25,50 } { 50 } { 25 } empty What’s in your wallet?! Lara Pudwell

  33. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chains for coins In the land of simplicity there are 25-cent and 50-cent coins. All prices end in 0, 25, 50, or 75 cents. Possible wallet states? charged 0 25 50 75 start empty empty { 25,50 } { 50 } { 25 } { 25 } { 25 } empty { 25,50 } { 25,25 } { 50 } { 50 } { 25 } empty { 25,50 } { 25,25 } { 25,25 } { 25 } empty { 25,25,25 } { 25,50 } { 25,50 } { 50 } { 25 } empty What’s in your wallet?! Lara Pudwell

  34. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chains for coins In the land of simplicity there are 25-cent and 50-cent coins. All prices end in 0, 25, 50, or 75 cents. Possible wallet states? charged 0 25 50 75 start empty empty { 25,50 } { 50 } { 25 } { 25 } { 25 } empty { 25,50 } { 25,25 } { 50 } { 50 } { 25 } empty { 25,50 } { 25,25 } { 25,25 } { 25 } empty { 25,25,25 } { 25,50 } { 25,50 } { 50 } { 25 } empty What’s in your wallet?! Lara Pudwell

  35. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chains for coins In the land of simplicity there are 25-cent and 50-cent coins. All prices end in 0, 25, 50, or 75 cents. Possible wallet states? charged 0 25 50 75 start empty empty { 25,50 } { 50 } { 25 } { 25 } { 25 } empty { 25,50 } { 25,25 } { 50 } { 50 } { 25 } empty { 25,50 } { 25,25 } { 25,25 } { 25 } empty { 25,25,25 } { 25,50 } { 25,50 } { 50 } { 25 } empty What’s in your wallet?! Lara Pudwell

  36. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chains for coins In the land of simplicity there are 25-cent and 50-cent coins. All prices end in 0, 25, 50, or 75 cents. Possible wallet states? charged 0 25 50 75 start empty empty { 25,50 } { 50 } { 25 } { 25 } { 25 } empty { 25,50 } { 25,25 } { 50 } { 50 } { 25 } empty { 25,50 } { 25,25 } { 25,25 } { 25 } empty { 25,25,25 } { 25,50 } { 25,50 } { 50 } { 25 } empty { 25,25,25 } { 25,25,25 } { 25,25 } { 25 } empty What’s in your wallet?! Lara Pudwell

  37. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Markov chains for coins In the land of simplicity there are 25-cent and 50-cent coins. All prices end in 0, 25, 50, or 75 cents. Possible wallet states? charged 0 25 50 75 start empty empty { 25,50 } { 50 } { 25 } { 25 } { 25 } empty { 25,50 } { 25,25 } { 50 } { 50 } { 25 } empty { 25,50 } { 25,25 } { 25,25 } { 25 } empty { 25,25,25 } { 25,50 } { 25,50 } { 50 } { 25 } empty { 25,25,25 } { 25,25,25 } { 25,25 } { 25 } empty What’s in your wallet?! Lara Pudwell

  38. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Simplicity wallet states (empty), (25), (50), (25, 25), (50, 25), (25, 25, 25)   1 / 4 1 / 4 1 / 4 0 1 / 4 0 1 / 4 1 / 4 0 1 / 4 1 / 4 0     1 / 4 1 / 4 1 / 4 0 1 / 4 0   P =   1 / 4 1 / 4 0 1 / 4 0 1 / 4     1 / 4 1 / 4 1 / 4 0 1 / 4 0   1 / 4 1 / 4 0 1 / 4 0 1 / 4 What’s in your wallet?! Lara Pudwell

  39. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Simplicity in the long run � � We want: L = p ( empty ) p (25) p (50) p (25 , 25) p (50 , 25) p (25 , 25 , 25) Setup: LP = L What’s in your wallet?! Lara Pudwell

  40. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Simplicity in the long run � � We want: L = p ( empty ) p (25) p (50) p (25 , 25) p (50 , 25) p (25 , 25 , 25) Setup: LP = L p ( empty ) p (25) p (50) p (25 , 25) p (50 , 25) p (25 , 25 , 25) + + + + + = p ( empty ) 4 4 4 4 4 4 p ( empty ) p (25) p (50) p (25 , 25) p (50 , 25) p (25 , 25 , 25) + + + + + = p (25) 4 4 4 4 4 4 p ( empty ) p (50) p (50 , 25) + + = p (50) 4 4 4 p (25) p (25 , 25) p (25 , 25 , 25) + + = p (25 , 25) 4 4 4 p ( empty ) p (25) p (50) p (50 , 25) + + + = p (50 , 25) 4 4 4 4 p (25 , 25) p (25 , 25 , 25) + = p (25 , 25 , 25) 4 4 p ( empty ) + p (25) + p (50) + p (25 , 25) + p (50 , 25) + p (25 , 25 , 25) =1 What’s in your wallet?! Lara Pudwell

  41. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Simplicity in the long run � � We want: L = p ( empty ) p (25) p (50) p (25 , 25) p (50 , 25) p (25 , 25 , 25) Setup: LP = L p ( empty ) p (25) p (50) p (25 , 25) p (50 , 25) p (25 , 25 , 25) + + + + + = p ( empty ) 4 4 4 4 4 4 p ( empty ) p (25) p (50) p (25 , 25) p (50 , 25) p (25 , 25 , 25) + + + + + = p (25) 4 4 4 4 4 4 p ( empty ) p (50) p (50 , 25) + + = p (50) 4 4 4 p (25) p (25 , 25) p (25 , 25 , 25) + + = p (25 , 25) 4 4 4 p ( empty ) p (25) p (50) p (50 , 25) + + + = p (50 , 25) 4 4 4 4 p (25 , 25) p (25 , 25 , 25) + = p (25 , 25 , 25) 4 4 p ( empty ) + p (25) + p (50) + p (25 , 25) + p (50 , 25) + p (25 , 25 , 25) =1 � 1 1 5 3 7 1 � Solve to get: 4 4 32 32 32 32 � � or 0 . 25 0 . 25 0 . 15625 0 . 09375 0 . 21875 0 . 03125 What’s in your wallet?! Lara Pudwell

  42. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun The Coin Keeper 1 The Simple Spender 2 The Gamblers 3 The Small Spender 4 The Whole Shebang 5 ...and More Fun 6 What’s in your wallet?! Lara Pudwell

  43. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun More assumptions 1 The fractional parts of prices are distributed uniformly between 0 and 99 cents. 2 Cashiers return change using the greedy algorithm. What’s in your wallet?! Lara Pudwell

  44. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun More assumptions 1 The fractional parts of prices are distributed uniformly between 0 and 99 cents. 2 Cashiers return change using the greedy algorithm. 3 If a spender does not have sufficient change to pay for their purchase, the spender spends no coins (and receives change from the cashier). What’s in your wallet?! Lara Pudwell

  45. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun More assumptions 1 The fractional parts of prices are distributed uniformly between 0 and 99 cents. 2 Cashiers return change using the greedy algorithm. 3 If a spender does not have sufficient change to pay for their purchase, the spender spends no coins (and receives change from the cashier). 4 If a spender has sufficient change, he or she makes their purchase by over-paying as little as possible (and receives change if necessary). What’s in your wallet?! Lara Pudwell

  46. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun More assumptions 1 The fractional parts of prices are distributed uniformly between 0 and 99 cents. 2 Cashiers return change using the greedy algorithm. 3 If a spender does not have sufficient change to pay for their purchase, the spender spends no coins (and receives change from the cashier). 4 If a spender has sufficient change, he or she makes their purchase by over-paying as little as possible (and receives change if necessary). 5 If there are multiple ways to overpay as little as possible, the spender favors spending a bigger coin over a smaller coin. What’s in your wallet?! Lara Pudwell

  47. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun What’s (the most) in your wallet? If you have at most 99 cents before a transaction, you’ll have at most 99 cents after. ◮ Case 1: (price ≤ wallet): You pay, and have less money in your wallet. What’s in your wallet?! Lara Pudwell

  48. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun What’s (the most) in your wallet? If you have at most 99 cents before a transaction, you’ll have at most 99 cents after. ◮ Case 1: (price ≤ wallet): You pay, and have less money in your wallet. ◮ Case 2: (price > wallet): You get (100 − p ) in change, and end up with (100 − p ) + w = 100 − ( p − w ) < 100. What’s in your wallet?! Lara Pudwell

  49. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Is this Markov chain regular? What’s in your wallet?! Lara Pudwell

  50. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Is this Markov chain regular? Yes! What’s in your wallet?! Lara Pudwell

  51. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Is this Markov chain regular? Yes! To get from any wallet to the empty wallet, imagine you have exact change. What’s in your wallet?! Lara Pudwell

  52. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Is this Markov chain regular? Yes! To get from any wallet to the empty wallet, imagine you have exact change. To get from empty wallet to { p pennies , n nickels , d dimes , q quarters } , imagine: q 75 cent charges d 90 cent charges n 95 cent charges p 99 cent charges What’s in your wallet?! Lara Pudwell

  53. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Counting states Known: Must have at most 99 cents. In other words, at most... 99 pennies 19 nickels 9 dimes 3 quarters 100 × 20 × 10 × 4 = 80 , 000 possible states. ... but that’s overkill. There are 6720 combinations of coins with at most 99 cents. What’s in your wallet?! Lara Pudwell

  54. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun That’s one big matrix... Goal: Find L where LP = L . What’s in your wallet?! Lara Pudwell

  55. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun 25 CPU hours later... p state p state Wallet state Wallet state 0 pennies .01000 { 25, 1, 1, 1 } .00453 1 penny .01000 { 5, 1, 1, 1 } .00448 2 pennies .01000 { 10, 5, 1, 1, 1, 1 } .00439 3 pennies .01000 { 25, 1, 1 } .00429 4 pennies .01000 { 10, 1, 1, 1 } .00420 5 pennies .00813 { 10, 1, 1, 1, 1, 1 } .00414 6 pennies .00732 { 25, 1 } .00405 7 pennies .00644 { 10, 1, 1, 1, 1, 1, 1 } .00391 8 pennies .00551 { 25 } .00379 { 5, 1, 1, 1, 1 } .00543 { 10, 5, 1, 1, 1, 1, 1, 1 } .00377 { 25, 1, 1, 1, 1 } .00475 { 25, 1, 1, 1, 1, 1 } .00376 { 10, 1, 1, 1, 1 } .00467 { 10, 5, 1, 1, 1, 1, 1 } .00375 9 pennies .00456 { 5, 1, 1, 1, 1, 1 } .00374 What’s in your wallet?! Lara Pudwell

  56. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun In case you were wondering... Expected number of coins in your wallet: 10.04 ◮ Expected number of quarters: 1.06 (10.6%) ◮ Expected number of dimes: 1.15 (11.4%) ◮ Expected number of nickels: 0.91 (9.1%) ◮ Expected number of pennies: 6.92 (68.9%) What’s in your wallet?! Lara Pudwell

  57. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun In case you were wondering... Expected number of coins in your wallet: 10.04 ◮ Expected number of quarters: 1.06 (10.6%) ◮ Expected number of dimes: 1.15 (11.4%) ◮ Expected number of nickels: 0.91 (9.1%) ◮ Expected number of pennies: 6.92 (68.9%) Probability of empty wallet: 0.01 Probability of having at least one nickel: 0.58085 Probability of having at least one penny: 0.95975 Probability of having only pennies (and a non-empty wallet): 0.08430 Probability of being able to pay any price with exact change: 0.00831 What’s in your wallet?! Lara Pudwell

  58. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun The Coin Keeper 1 The Simple Spender 2 The Gamblers 3 The Small Spender 4 The Whole Shebang 5 ...and More Fun 6 What’s in your wallet?! Lara Pudwell

  59. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun And that’s not all! Some (common?) variations The pennyless purchaser The quarter hoarder The pennies-first spender The Shallit currency Jeffrey Shallit, What this country needs is an 18 ¢ piece, The Mathematical Intelligencer 25 (2003) 20–23. What’s in your wallet?! Lara Pudwell

  60. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun And that’s not all! Some (common?) variations The pennyless purchaser (5, 10, and 25-cent pieces) The quarter hoarder (1, 5, and 10-cent pieces) The pennies-first spender (1, 5, 10, and 25-cent pieces) The Shallit currency (1, 5, 18, and 25-cent pieces) Jeffrey Shallit, What this country needs is an 18 ¢ piece, The Mathematical Intelligencer 25 (2003) 20–23. What’s in your wallet?! Lara Pudwell

  61. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun And that’s not all! Some (common?) variations The pennyless purchaser (5, 10, and 25-cent pieces) Go The quarter hoarder (1, 5, and 10-cent pieces) Go The pennies-first spender (1, 5, 10, and 25-cent pieces) Go The Shallit currency (1, 5, 18, and 25-cent pieces) Go Jeffrey Shallit, What this country needs is an 18 ¢ piece, The Mathematical Intelligencer 25 (2003) 20–23. Go What’s in your wallet?! Lara Pudwell

  62. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Pennyless purchaser 213 states What’s in your wallet?! Lara Pudwell

  63. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Pennyless purchaser results p pp p pp Wallet state Wallet state 1 . 29 × 10 − 11 {} .05000 14 nickels 3 . 37 × 10 − 12 { 5 } .05000 2 dimes and 15 nickels 2 . 28 × 10 − 12 { 10, 5 } .03916 1 dime and 15 nickels 9 . 90 × 10 − 13 { 25, 10, 5 } .03093 15 nickels 1 . 76 × 10 − 13 { 25, 5 } .02847 1 dime and 16 nickels 6 . 23 × 10 − 14 { 10, 5, 5 } .02731 16 nickels 1 . 27 × 10 − 14 { 25, 25, 10, 5 } .02625 1 dime and 17 nickels 3 . 96 × 10 − 15 { 5, 5 } .02536 17 nickels 2 . 09 × 10 − 16 { 10 } .02463 18 nickels 1 . 10 × 10 − 17 { 25, 10, 5, 5 } .02417 19 nickels Go What’s in your wallet?! Lara Pudwell

  64. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Quarter hoarder 4125 states What’s in your wallet?! Lara Pudwell

  65. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Quarter hoarder results p qh p qh Wallet state Wallet state { 1, 1, 1, 1 } .01164 10 pennies .00713 { 1, 1, 1 } .01129 { 10,5,1,1,1,1,1,1,1,1 } .00651 { 1, 1 } .01095 { 10, 5, 1, 1, 1, 1, 1, 1, 1 } .00642 5 pennies .01084 11 pennies .00638 { 1 } .01062 { 10, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1 } .00637 6 pennies .01039 { 10, 5, 1, 1, 1, 1, 1, 1 } .00614 {} .01030 { 10, 5, 1, 1, 1, 1, 1 } .00569 7 pennies .00984 12 pennies .00564 8 pennies .00919 { 10, 5, 1, 1, 1, 1 } .00549 9 pennies .00844 { 10, 1, 1, 1, 1, 1, 1 } .00523 Go What’s in your wallet?! Lara Pudwell

  66. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Pennies-first spender 1065 states What’s in your wallet?! Lara Pudwell

  67. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Pennies-first results Expected pennies-first coins in your Expected number of coins in your wallet: 5.74 wallet: 10.04 Expected quarters: 1.12 Expected quarters: 1.06 Expected dimes: 1.27 Expected dimes: 1.15 Expected nickels: 1.35 Expected nickels: 0.91 Expected pennies: 2.00 Expected pennies: 6.92 Go What’s in your wallet?! Lara Pudwell

  68. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun The Shallit currency Idea: replacing a dime with an 18-cent coin minimizes coins used per transaction Two catches: Greedy algorithm isn’t always best! Example: 28 cents Greedy: 25+1+1+1 Efficient: 18+5+5 What’s in your wallet?! Lara Pudwell

  69. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun The Shallit currency Idea: replacing a dime with an 18-cent coin minimizes coins used per transaction Two catches: Greedy algorithm isn’t always best! Example: 28 cents Greedy: 25+1+1+1 Efficient: 18+5+5 There isn’t always a unique way to give the fewest possible coins! Example: 77 cents 25 + 25 + 25 + 1 + 1 = 77 18 + 18 + 18 + 18 + 5 = 77 What’s in your wallet?! Lara Pudwell

  70. The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun The Shallit currency Idea: replacing a dime with an 18-cent coin minimizes coins used per transaction Two catches: Greedy algorithm isn’t always best! Example: 28 cents Greedy: 25+1+1+1 Efficient: 18+5+5 There isn’t always a unique way to give the fewest possible coins! Example: 77 cents 25 + 25 + 25 + 1 + 1 = 77 18 + 18 + 18 + 18 + 5 = 77 Assumptions: Spenders: still break ties by using bigger coins. Cashiers: break ties by using each “best” change equally often. What’s in your wallet?! Lara Pudwell

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