Combinations and State Diagrams Discrete Structures CS 173

Combinations and State Diagrams Discrete Structures (CS 173) Derek Hoiem, University of Illinois 04/16/13 1

HW 9 Do not use the quadratic formula! Review set constructor notation from earlier chapter. Relations between sets that we’ve covered include equals, subset, proper subset, disjoint. 2

Today’s class • Counting and combinations • State diagrams (mainly applied to counting) 3

Combinations/counting • 4

Mixed combination problems Examples • Suppose a slot machine has 6 dials which can each be set to {bell, cherry, 777, blank 1, blank 2} – How many possible (ordered) combinations are there? – How many ways are there to get exactly three cherries? – How many ways are there to get at least three cherries? 8

Dice games When rolling five dice at once, which is more likely, three-of-a-kind or a large-straight? 11

Binomial theorem • 12

Binomial theorem • 13

State diagrams state transition action 14

State diagrams and counting Suppose you have 3 red pills and 2 blue pills in a pouch. You draw three of them from at random. What is the probability that you have exactly one blue pill? 15

Dice games • 16

Challenge problem If Joe is going for “Yahtzee” (five of a kind with five dice), what is the chance that he will get it within two re-rolls? 17

Suppose we roll a two-sided die until the sum is a non-zero multiple of 3. What is the chance of getting there in three or fewer rolls? 18

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Things to remember • 20

Next class More finite state machines 21