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COMBINATORICS AND PROBABILITY

Explore the basics and advanced principles of combinatorics, including permutations, combinations, factorials, and the multiplication and addition principles. Learn to apply these concepts in solving problems involving counting and probability.

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COMBINATORICS AND PROBABILITY

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  1. T BOLAN COMBINATORICSANDPROBABILITY menu

  2. MENU BASICS COMBINATORICS: COMBINATION / PERMUTATION PERMUTATIONS with REPETITION BINOMIAL THEOREM (light) PROBABILITY: BASICS OF EVENTS TOGETHER WITH COMBINATIONS menu

  3. COMBINATORICS BASICS Combinatorics means “ways of counting”. Of course that seems simple, but it is used to identify patterns and shortcuts for counting large numbers of possibilities menu

  4. ! ? ? COMBINATORICS BASICS The multiplication principle If an action can be performed n ways, and for each of those actions, a second action can be performed p ways, then the two actions together can be performed np ways Say you have 3 different pairs of socks, and 2 different pairs of shoes, how many ways can you put On footware? menu

  5. COMBINATORICS BASICS The multiplication principle Say you have 3 different pairs of socks, and 2 different pairs of shoes, how many ways can you put On footware? sox A sox B sox C shoes 1 shoes 2 shoes 1 shoes 2 shoes 1 shoes 2 1 2 3 4 5 6 3*2 = 6 Six ways to put on shoes and sox menu BACK SKIP

  6. ! ? ? COMBINATORICS BASICS The addition principle If two actions are mutually exclusive, and the first action can be done n ways, and the second can be done p ways, then one action OR the other can be done n + p ways. First, let’s explain this The key word in understanding “mutually exclusive” Is OR menu

  7. COMBINATORICS BASICS MUTUALLY EXCLUSIVE: Mutually exclusive basically means that two things CANNOT both occur. If the Cubs play the Sox, then there are 3 possible outcomes: Cubs win Sox win no one wins (rain out etc.) Can both of these happen in the same game? NO. they are mutually exclusive In other words, they cannot BOTH happen menu

  8. COMBINATORICS BASICS MUTUALLY EXCLUSIVE: Pick a number 0-9: You have 10 choices Pick a number A-Z: You have 26 choices Pick a letter OR a number: You have 10+26=36 choices Since you can not pick both, you add the number of options together. menu

  9. COMBINATORICS BASICS Factorials “five factorial” It means multiply the number by each integer smaller than it THIS WILL BE EXTREMELY USEFUL menu

  10. COMBINATORICS BASICS Factorials Just take my word on this one for now. menu

  11. COMBINATORICS BASICS Factorials menu

  12. COMBINATORICS BASICS Arrangements: That’s how many ways 5 people can be arranged You have 5 different people, How many ways can they line up? How many choices for Who is first? Now, how many choices for second? menu

  13. COMBINATORICS BASICS SIDE NOTE: How many ways can you arrange zero things? 1 By doing nothing. You have no choice. That’s why 0! = 1 menu

  14. COMBINATION / PERMUTAION To continue, we must understand the difference between COMBINATIONS and PERMUTATIONS ORDER DOESN’T MATTER ORDER MATTERS menu

  15. COMBINATION / PERMUTAION Out of a group of 10 people, you are going to pick 3 to win a new car. Each winner gets only 1 car PERMUTATION Pick the first winner. How many choices do you have? Rolls Royce 10 Cadillac Pick the second winner. How many choices do you have? 9 Yugo Pick the third winner. How many choices do you have? 8 menu

  16. COMBINATION / PERMUTAION Out of a group of 10 people, you are going to pick 3 to win a new car. Each winner gets only 1 car Yugo Pick the first winner. How many choices do you have? 10 Yugo Pick the second winner. How many choices do you have? 9 Yugo Pick the third winner. How many choices do you have? 8 menu

  17. COMBINATION / PERMUTAION Out of a group of 10 people, you are going to pick 3 to win a new car. Each winner gets only 1 car COMBINATION Yugo But in this case, since all the cars are the same, would it matter if you picked The green guy first and the purple guy second? Yugo NO. This gives the same results, each person Still got the exact same prize Yugo So we don’t count all the different ways To do this menu

  18. COMBINATION / PERMUTAION COMBINATIONS and PERMUTATIONS ORDER DOESN’T MATTER ORDER MATTERS All the prizes or slots or positions Are the same. All the prizes or slots or positions Are the different. ie, every winner gets the same thing ie, first place, second place etc menu

  19. COMBINATION / PERMUTAION If you have n objects to choose from In this case 10 people And you have r slots to put them in In this case 3 winners If order doesn’t matter (each slot is the same) If order does matter (each slot is different) COMBINATION PERMUTATION The extra r down here is to get rid of the extra ways to arrange the winners menu

  20. COMBINATION / PERMUTAION Out of a group of 10 people, you are going to pick 3 to win a new car. Each winner gets only 1 car Does order matter? NO So this is a COMBINATION menu

  21. COMBINATION / PERMUTAION Out of a group of 10 people, you are going to pick 3 to win a new car. Each winner gets only 1 car Does order matter? YES So this is a PERMUTATION menu

  22. COMBINATION / PERMUTAION In a class of 20 students, we will pick 3 to represent us On the student counsel Representatives (all the same job) 20 19 18 Bob Jamie Jim How many choices do we have for the first slot? 20 But, since all the positions are the same, Bob, Jamie, Jim is the same as Jim, Jamie, Bob How many choices do we have for the second slot? 19 So we have to divide by the number of repeats How many ways can I rearrange 3 people? How many choices do we have for the third slot? 18 3! = 6 menu

  23. COMBINATION / PERMUTAION In a class of 20 students, we will pick 3 to represent us On the student counsel Representatives (all the same job) Let’s see if this is any faster with the formula… menu

  24. COMBINATION / PERMUTAION In a class of 20 students, we will pick 3 to represent us On the student counsel President, Vice President, Secretary 20 19 18 SO… Ways to do this Bob Jamie Jim How many choices do we have for the first slot? 20 OR THIS WAY How many choices do we have for the second slot? 19 How many choices do we have for the third slot? 18 menu

  25. PERMUTATIONS with REPETITION How many ways can you rearrange the letters of the word CAT? Three letters, and three slots to put them in. 3 2 1 CAT CTAACTATCTACTCA Three choices for the first letter. Two choices for the second letter. One choice for the third letter. menu

  26. PERMUTATIONS with REPETITION How many ways can you rearrange the letters of the word MOM? Three letters, and three slots to put them in. 3 2 1 MOM OMMMMO What went wrong? We can’t really count the 2 m’s as separate letters. Starting with the first m or second m gives the same results! menu

  27. PERMUTATIONS with REPETITION How many ways can you rearrange the letters of the word MOM? So for mom, 3 letters, O appears once M appears twice: The number of letters in the word Yes, you can ignore the 1!’s How many times each letter appears menu

  28. PERMUTATIONS with REPETITION How many ways can you rearrange the letters of the word BOOKKEEPER? 10 letters, O appears twice K appears twice E appears three times menu

  29. PERMUTATIONS with REPETITION How many ways can you sit 4 people at a circular table? Before you say it is 4!, aren’t these the same? Jack Izzy Izzy Brooke Rebecca Jack Jack Brooke Rebecca Izzy Rebecca Brooke All we’ve really done is rotate the table. menu

  30. PERMUTATIONS with REPETITION How many ways can you sit 4 people at a circular table? It is 4! But, 4 of the arrangements are the same! However many “people” there are, There will be the same number of repeats Jack Brooke Rebecca Izzy SHORTCUT: menu

  31. BINOMIAL THEOREM See if you can figure out the pattern… Each number is the sum of the two above it. 1 1 1 2 1 1 3 3 1 This is called PASCAL’s TRIANGLE 1 4 6 4 1 1 5 10 10 5 1 Named for Blaise Pascal 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 menu

  32. BINOMIAL THEOREM Now here’s something kinda strange… 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 …And so on, and so on. 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 menu

  33. BINOMIAL THEOREM Now try these: 1 1 1 1 1 2 1 1 2 1 1 3 3 1 1 3 3 1 1 4 6 4 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 menu

  34. PROBABILITY BASICS Any action with unpredictable outcomes EXPERIMENT: SAMPLE SPACE: The collection of all possible outcomes EVENT: A specific outcome P (A): The probability of outcome A occurring menu

  35. PROBABILITY BASICS Probabilities are given in decimals, fractions or percents The probabilities of all possible outcomes should always add to 1 or 100% In other words: If there is a 40% chance it will rain tomorrow, What is the probability it will NOT rain? 60% OR 0.4 and 0.6 OR 2/5 and 3/5 menu

  36. ! ? ? A B PROBABILITY BASICS PROBABILITY OF 2 EVENTS For any 2 events A and B, P(A or B) = P(A) + P(B) - P(A and B) This is easy with a Venn Diagram: menu

  37. A B PROBABILITY BASICS For any 2 events A and B, P(A or B) = P(A) + P(B) - P(A and B) P(A) = 0.4 P(B) = 0.3 0.4 0.3 P(A and B) = 0.1 0.1 So if we wanted A or B, we add up their probabilities. This means that the probability of A is the entire circle A = 0.4 And the probability of B is the entire circle B = 0.3 But these 2 circles overlap, and if we want A or B, then that overlapping part might be counted twice! menu

  38. PROBABILITY BASICS Still not making sense? Try this… You want to know the weather tomorrow. cloudy 0.4 windy The chances it will be cloudy: P(C) = 0.4 0.3 0.1 The chances it will be Windy: P(W) = 0.3 What is the probability that it will be windy OR cloudy tomorrow? P(C or W) P(C) + P(W) = 0.4 + 0.3 = 0.7 P(C) + P(W) - P(W and C) = 0.4 + 0.3 - 0.1 =0.6 Why is this wrong? That middle 0.1 is being counted twice. So we must subtract it menu

  39. A B PROBABILITY BASICS PROBABILITY OF 2 EVENTS For any 2 events A and B, P(A or B) = P(A) + P(B) - P(A and B) This is easy with a Venn Diagram: menu

  40. PROBABILITY BASICS WHAT IF THEY ARE MUTUALLY ECLUSIVE? 0.4 For any 2 events A and B, that are mutually exclusive P(A or B) = P(A) + P(B) 0.3 For any 2 events A and B, P(A or B) = P(A) + P(B) - P(A and B) This is the overlap. But they don’t overlap, so it is 0 menu

  41. PROBABILITY BASICS MISC. stuff you have to know in probability. CARDS: there are 52 in a deck there are 4 suits: (hearts (red), diamonds (red), spades (black), clubs (black)) each suit has 13 cards: 2-10, Jack, Queen, King, Ace The Jack, Queen and King are called Face cards each card has an equal chance of being pulled. DICE: A regular die has 6 sides. Each side has a number (1-6) each side has an equal chance of being rolled. menu

  42. PROBABILITY BASICS BASIC CALCULATIONS What is the probability of drawing... # of ways this outcome can happen ie how many diamonds there are The king of hearts? The 10 of diamonds? Any diamond? Any king? An even numbered card? Any red card? A four? Any face card? Not a diamond? How many total possible outcomes there are. (there are 52 possible different draws) menu

  43. PROBABILITY BASICS BASIC CALCULATIONS What is the probability of drawing... The king of hearts? The 10 of diamonds? Any diamond? Any king? An even numbered card? Any red card? menu

  44. PROBABILITY BASICS BASIC CALCULATIONS What is the probability of drawing... A four? Any face card? Not a diamond? menu

  45. PROBABILITY BASICS IMPORTANT MISC. What is the probability of drawing... All draws fall into one of these two categories, so what should their probabilities add to? The king of hearts? The 10 of diamonds? Any diamond? Any king? An even numbered card? Any red card? A four? Any face card? Not a diamond? menu

  46. PROBABILITY BASICS IMPORTANT MISC. What is the probability of drawing an object with mutually exclusive properties? For example a black diamond? 0. There is no such thing. menu

  47. EVENTS OCCURING TOGETHER What is the probability of flipping a coin and getting a heads then a tails. Though you don’t have to do the problem this way, we will start this one by listing all the different possibilities: T H First flip: H T H T Second flip: H, HH, T T, H T, T menu

  48. EVENTS OCCURING TOGETHER What is the probability of flipping a coin and getting a heads then a tails. Though you don’t have to do the problem this way, we will start this one by listing all the different possibilities: T H First flip: H T H T Second flip: H, HH, T T, H T, T menu

  49. EVENTS OCCURING TOGETHER What is the probability of flipping a coin and getting a heads then a tails. Though you don’t have to do the problem this way, we will start this one by listing all the different possibilities: T H First flip: H T H T Second flip: H, HH, T T, H T, T menu

  50. EVENTS OCCURING TOGETHER What is the probability of flipping a coin and getting a heads then a tails. Though you don’t have to do the problem this way, we will start this one by listing all the different possibilities: T H First flip: H T H T Second flip: H, HH, T T, H T, T menu

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