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COMBINATORICS AND PROBABILITY T BOLAN menu. MENU COMBINATORICS: BASICS COMBINATION / PERMUTATION PERMUTATIONS with REPETITION BINOMIAL THEOREM (light)

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Presentation on theme: "COMBINATORICS AND PROBABILITY T BOLAN menu. MENU COMBINATORICS: BASICS COMBINATION / PERMUTATION PERMUTATIONS with REPETITION BINOMIAL THEOREM (light)"— Presentation transcript:

1 COMBINATORICS AND PROBABILITY T BOLAN menu

2 MENU COMBINATORICS: BASICS 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 shoes 1shoes 2 sox B shoes 1shoes 2 sox C shoes 1shoes 2 123456 3*2 = 6Six ways to put on shoes and sox BACKSKIP menu

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: 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? That’s how many ways 5 people can be arranged menu

13 COMBINATORICS BASICS SIDE NOTE: How many ways can you arrange zero things? 1By 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 MATTERSORDER DOESN’T MATTER 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 Rolls Royce Cadillac Yugo Pick the first winner. How many choices do you have? 10 Pick the second winner. How many choices do you have? 9 Pick the third winner. How many choices do you have? 8 PERMUTATION 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 Pick the second winner. How many choices do you have? 9 Pick the third winner. How many choices do you have? 8 Yugo 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 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 So we don’t count all the different ways To do this COMBINATION menu

18 COMBINATION / PERMUTAION COMBINATIONS and PERMUTATIONS ORDER MATTERSORDER DOESN’T MATTER 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 And you have r slots to put them in In this case 10 people In this case 3 winners If order doesn’t matter (each slot is the same) If order does matter (each slot is different) COMBINATIONPERMUTATION 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? NOSo 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? YESSo 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) How many choices do we have for the first slot? 20 How many choices do we have for the second slot? 19 How many choices do we have for the third slot? 18 201918 But, since all the positions are the same, BobJamieJim Bob, Jamie, Jim is the same as Jim, Jamie, Bob So we have to divide by the number of repeats How many ways can I rearrange 3 people? 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 How many choices do we have for the first slot? 20 How many choices do we have for the second slot? 19 How many choices do we have for the third slot? 18 201918 SO…Ways to do this BobJamieJim OR THIS WAY 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. Three choices for the first letter. Two choices for the second letter. One choice for the third letter. 321 CAT CTA ACT ATC TAC TCA 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. 321 MOM OMM MMO 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? The number of letters in the word How many times each letter appears So for mom, 3 letters, O appears once M appears twice: Yes, you can ignore the 1!’s 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? Jack Brooke Rebecca Izzy Before you say it is 4!, aren’t these the same? Jack Brooke Rebecca Izzy Jack Brooke Rebecca Izzy 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? Jack Brooke Rebecca Izzy It is 4! But, 4 of the arrangements are the same! However many “people” there are, There will be the same number of repeats SHORTCUT: menu

31 BINOMIAL THEOREM See if you can figure out the pattern… 1 1 2 1 1 3 3 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 Each number is the sum of the two above it. This is called PASCAL’s TRIANGLE Named for Blaise Pascal menu

32 BINOMIAL THEOREM Now here’s something kinda strange… 1 1 2 1 1 3 3 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 …And so on, and so on. menu

33 BINOMIAL THEOREM Now try these: 1 1 2 1 1 3 3 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 11 121 1331 14641 menu

34 PROBABILITY BASICS EXPERIMENT: SAMPLE SPACE: EVENT: Any action with unpredictable outcomes The collection of all possible outcomes 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 0.6OR 2/5 and 3/5 menu

36 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: A B menu

37 PROBABILITY BASICS For any 2 events A and B, P(A or B) = P(A) + P(B) - P(A and B) A B P(A) = 0.4 P(B) = 0.3 P(A and B) = 0.1 0.4 0.3 0.1 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 So if we wanted A or B, we add up their probabilities. 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… cloudy windy0.4 0.3 0.1 You want to know the weather tomorrow. The chances it will be cloudy: P(C) = 0.4 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 Why is this wrong? That middle 0.1 is being counted twice. So we must subtract it P(C) + P(W) - P(W and C) = 0.4 + 0.3 - 0.1 =0.6 menu

39 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: A B menu

40 PROBABILITY BASICS WHAT IF THEY ARE MUTUALLY ECLUSIVE? 0.4 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 For any 2 events A and B, that are mutually exclusive P(A or B) = P(A) + P(B) 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... 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? # of ways this outcome can happen ie how many diamonds there are 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... 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? All draws fall into one of these two categories, so what should their probabilities add to? 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: First flip: H T Second flip: HHTT H, H H, 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: First flip: H T Second flip: HHTT H, H H, 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: First flip: H T Second flip: HHTT H, H H, 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: First flip: H T Second flip: HHTT H, H H, T T, H T, T menu

51 EVENTS OCCURING TOGETHER What is the probability of flipping a coin and getting a heads then a tails. H T HHTT “TREE DIAGRAM” There are four different outcomes, how many of them are “good”? menu

52 EVENTS OCCURING TOGETHER When we talk about 2 things happening, one after the other, there are 2 possibilities: Either the first thing affects the second, or The first thing does NOT affect the second CONDITIONAL INDEPENDANT menu

53 EVENTS OCCURING TOGETHER If you know the probabilities along the “tree diagram”, you can find the probability of a specific outcome by multiplying all the probabilities along the path to that outcome. H T HHTT The probability of getting “H” then “T” is menu

54 EVENTS OCCURING TOGETHER The first affects the second: CONDITIONAL A jar is filled with 5 red, and 5 green Jelly Beans. What are the chances of randomly pulling out a green one? After you have pulled out and eaten that green one, what are the chances of randomly pulling out a second green one? four green nine total five green ten total But the probability of the second draw would be much different if.. I asked you to draw a RED one then a GREEN one 0.555 So the first draw affected the odds in the second draw. THAT is what conditional probability means menu

55 EVENTS OCCURING TOGETHER If the probability of “B” is conditional (affected by) “A” (what happens before it) then we say: The probability of “B” given ”A” P(B|A) The probability that this will happen Assuming this already HAS happened menu

56 EVENTS OCCURING TOGETHER In a jar are 5 RED and 5 GREEN jelly beans. What is the probability that the second Jelly bean drawn from the jar is GREEN given that the first jelly bean is RED? P(G|R) The second JB is GREENIf the first one is RED Starting off there are 5 RED and 5 Green jelly beans (10 total) After taking out one red, there are 4 RED and 5 GREEN (9 total) P(G|R)= menu

57 EVENTS OCCURING TOGETHER THE PROBABILITY OF 2 EVENTS A and B OCCURRING TOGETHER IS: If the events are independent If the events are conditional Probability of A times Probability of B Probability of A times Probability of B given A menu

58 EVENTS OCCURING TOGETHER Before we try some problems, there is one more thing you need to know: The probability of something happening that has already happened is always 1. EXAMPLE: What is the probability that man will walk on the moon? 1or 100% It already happened. menu

59 EVENTS OCCURING TOGETHER PRACTICE PROBLEMS What is the probability of drawing twice from a deck and getting the king of diamonds and the queen of diamonds in any order? menu

60 EVENTS OCCURING TOGETHER PRACTICE PROBLEMS What is the probability of drawing twice from a deck and getting the king of diamonds then the queen of diamonds? menu

61 EVENTS OCCURING TOGETHER PRACTICE PROBLEMS What is the probability of drawing twice from a deck and getting the king of diamonds then putting the card back in, then drawing the queen of diamonds? THIS IS CALLED REPLACEMENT menu

62 EVENTS OCCURING TOGETHER PRACTICE PROBLEMS When rolling a die, what is the probability of rolling a 5? When rolling a die, what is the probability of rolling an odd number? menu

63 EVENTS OCCURING TOGETHER PRACTICE PROBLEMS When rolling two die, what is the probability of rolling a 5? What are all the possibilities? There are 36 total possibilities menu

64 PROBABILITY with COMBINATIONS From a deck of 52 cards, what is the probability of drawing 5 hearts? How many ways can you draw 5 cards? How many ways can you get 5 hearts? menu

65 PROBABILITY with COMBINATIONS For all probability problems that are solved using combinations: # of ways the desired outcome can happen Total number of outcomes menu

66 PROBABILITY with COMBINATIONS When drawing from a deck of cards, what is the possibility of drawing a pair of aces? 4 aces, pick any 2 Total number of possible hands menu

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