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Probability, Permutations, Pizza, and Plinko !

Probability, Permutations, Pizza, and Plinko !. Klay Kruczek Southern Connecticut State University kruczekk3@southernct.edu. Discrete Mathematics. Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values.

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Probability, Permutations, Pizza, and Plinko !

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  1. Probability, Permutations, Pizza, and Plinko! KlayKruczek Southern Connecticut State University kruczekk3@southernct.edu

  2. Discrete Mathematics • Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. • The study of how discrete objects combine with one another and the probabilities of various outcomes is known as combinatorics. Other fields of mathematics that are considered to be part of discrete mathematics include graph theory. • The study of topics in discrete mathematics usually includes the study of algorithms, their implementations, and efficiencies. Discrete mathematics is the mathematical language of computer science. • http://mathworld.wolfram.com/DiscreteMathematics.html

  3. Discrete Mathematics • Discrete mathematics is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the notion of continuity. ...en.wikipedia.org/wiki/Discrete_mathematics • For me, discrete mathematics is about counting, playing games, coloring, and solving puzzles. – K

  4. Notes to Students • I cannot emphasize the following statements enough: ``The solutions of discrete mathematics problems can often be obtained using ad hoc arguments, possibly coupled with use of general theory. One cannot always fall back on application of formulas or known results'' To me, this means you have to play around sometimes. The answer is not always right in front of your face. I got into mathematics because I love solving puzzles. Discrete mathematics is that for me.

  5. Notes to Students To solve a discrete mathematics problem, you may have to perform the following steps: • Make sure you understand what the question is asking. • Determine what AN example of an object satisfying the requirements of the problem might look like. • Do some computations for small cases in order to develop an idea of what exactly is going on. (This is probably most important) • Try to develop some sort of systematic approach when doing small examples. • Use reasoning and possibly creativity to obtain a solution to the problem. • Spend more than 30 minutes trying to solve a problem.

  6. Counting and Counting Principles • Look at pages 1 – 3 in your packet; paying particular attention to #2, 3, 6– 8, 10, 11. • 2) 6 x 4…one can list the combinations • 3) You win $1 when the roll is from the set: • {(1,1), (1,2), (2,1), (2,2), (2,3), (3,2), (3,3), (3,4), (4,3), (4,4), (4,5)} (the first # listed is from the 4-sided die) • This is 11 times out of 24, so you lose one dollar 13 times out of 24, i.e., the game is not fair • 6) 63 games 7) N – 1 games 8) P(all right) =

  7. Counting and Counting Principles • 10) • 36 match-ups • 9 x 8 = 72 • 9 x 8 = 72 • 9 + 8 = 17

  8. Counting Principles Counting Principles • Suppose you have N1 objects in set S1, N2 objects in set S2 and N3 objects in set S3. • How many different ways are there to choose one object from either S1, S2 or S3? + • How many different ways are there to choose one object from S1 and one from S2 and one from S3?

  9. Permutations and Combinations with Ice Cream • Look at pages 4 – 5 in the packet • On page 4, the goal is to get students to notice, using the Multiplication Counting Principle, that • If you have n flavors of ice cream and you want to make an ice cream cone with k flavors (where order matters), then the number of possible cones is: • , • where is the number of permutations(order matters) of objects taken at a time

  10. Permutations and Combinations with Ice Cream • On page 5, students should be listing the possible ice cream cones (since order does not matter). • Possible student observations if making cones with k of n flavors: • (# when order matters) / (# when order does not matter) = k! • The # of cones you can make with 3 flavors is the same as 4 flavors, similarly for 1 and 6, 2 and 5, as well as 0 and 7. • Using the first observation above, we get • , where is the number of combinations (order does not matter) of objects taken at a time. • This gives

  11. Pizzas and Pascal’s Triangle • Look at pages 9 – 10 • Questions to ask before talking about pizza. • 5 toppings available: 32 pizzas possible (One 0-topping, Five 1-topping, Ten 2-topping, Ten 3-topping, Five 4-topping, One 5-topping) • 4 toppings available: 16 pizzas possible (One 0-topping, Four 1-topping, Six 2-topping, Four 3-topping, One 4-topping) • 3 toppings available: 8pizzas possible (One 0-topping, Three 1-topping, Three 2-topping, One 3-topping) • N toppings available: pizzas possible • N toppings available; k topping pizzas: pizzas possible

  12. Pizzas and Pascal’s Triangle The Connecdtion

  13. Probability and Powerball • “Lottery: A tax on ignorance” • Look at page 16 • 1/39 • (They pay you $10,000) • You can share the prize.

  14. Plinko! • Look at pages 17 – 18 • ● • ● ● • ● ● ● • ● ● ● ● • ● ● ● ● ● • A B C D E

  15. Plinko! • Look at pages 17 – 18 • ● 1 • ● ● ½ ½ • ● ● ● • ● ● ● ● • ● ● ● ● ● • A B C D E

  16. Plinko! • Look at pages 17 – 18 • ● 1 • ● ● ½ ½ • ● ● ● ¼ 2/4 ½ • ● ● ● ● • ● ● ● ● ● • A B C D E

  17. Plinko! • Look at pages 17 – 18 • ● 1 • ● ● ½ ½ • ● ● ● ¼ 2/4 ½ • ● ● ● ● 1/8 3/8 3/8 1/8 • ● ● ● ● ● • A B C D E

  18. Plinko! • Look at pages 17 – 18 • ● 1 • ● ● ½ ½ • ● ● ● ¼ 2/4 ½ • ● ● ● ● 1/8 3/8 3/8 1/8 • ● ● ● ● ● 1/16 4/16 6/16 4/16 1/16 • A B C D E A B C D E

  19. Plinko! • ● • ● ● • ● ● ● • ● ● ● ● • ● ● ● ● ● 1/16 4/16 6/16 4/16 1/16 • ● ● ● ● 3/16 5/16 5/16 3/16 • ● ● ● 11/32 5/16 11/32 • A B C A B C

  20. Pizza Delivery Problem • Pages 19 – 20: • Origin: Konigsberg Bridge Problem

  21. Pizza Delivery

  22. Pizza Delivery You either (i) start in Queens and in Brooklyn or (ii) start in Brooklyn and end in Queens

  23. Euler Circuits and Trails • When you start from a vertex in a graph, walk along every edge exactly once, and return to the starting vertex, we say this graph has an Euler Circuit. If you walk along every edge, but do not return to the starting vertex, we say the graph has an Euler Trail. • A graph has an Euler Circuit when the degree of each vertex is even (the # of edges leaving each vertex is even). • (II, IV, VI, VII, XII, XIII, XVI) • A graph has an Euler Trailwhen the graph has exactly two vertices of odd degree. • (I, V, VIII, XI, XIV)

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