T BOLAN COMBINATORICS AND PROBABILITY menu MENU COMBINATORICS

T BOLAN COMBINATORICS AND PROBABILITY menu

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

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

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

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 1 1 sox B shoes 2 shoes 1 shoes 2 3 4 5 6 2 3*2 = 6 menu sox C BACK Six ways to put on shoes and sox SKIP

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

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: Can both of these happen Cubs win in the same game? Sox win no one wins (rain out etc. ) NO. they are mutually exclusive In other words, they cannot BOTH happen menu

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

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

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

COMBINATORICS BASICS Factorials menu

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

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

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

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 menu Rolls Royce Pick the first winner. How many choices do you have? 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

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 menu 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

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

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

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 If order does matter (each slot is the same) (each slot is different) COMBINATION PERMUTATION The extra r down here is to get rid of the extra ways to arrange the winners menu

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 menu So this is a COMBINATION

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 menu So this is a PERMUTATION

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 How many choices do we have for the second slot? 19 How many choices do we have for the third slot? 18 menu But, since all the positions are the same, 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

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

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 How many choices do we have for the second slot? 19 How many choices do we have for the third slot? 18 menu OR THIS WAY

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 Three choices for the first letter. Two choices for the second letter. One choice for the third letter. menu 2 1 CAT CTA ACT ATC TAC TCA

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 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 1 MOM OMM MMO

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

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

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 Jack Brooke Rebecca Izzy menu Izzy Brooke Rebecca Jack Brooke All we’ve really done is rotate the table.

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! Jack Brooke Rebecca Izzy menu However many “people” there are, There will be the same number of repeats SHORTCUT:

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 1 menu 1 4 5 10 10 6 7 15 21 6 4 20 35 35 This is called PASCAL’s TRIANGLE 1 5 1 15 6 21 Named for Blaise Pascal 1 7 1

BINOMIAL THEOREM Now here’s something kinda strange… 11 1 2 1 1 3 3 1 1 1 4 5 10 10 6 7 menu 15 21 6 4 20 35 35 1 15 6 21 …And so on, and so on. 1 7 1

BINOMIAL THEOREM Now try these: 11 1 1 2 1 1 1 3 3 1 1 1 4 5 10 10 6 7 menu 15 21 6 4 20 35 1 15 6 21 7 1 2 3 4 1 1 1 3 6 1 4 1

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

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 menu OR 2/5 and 3/5

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 menu B

PROBABILITY BASICS For any 2 events A and B, P(A or B) = P(A) + P(B) - P(A and B) A 0. 4 0. 1 B 0. 3 P(A) = 0. 4 P(B) = 0. 3 P(A and B) = 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 menu 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!

PROBABILITY BASICS Still not making sense? Try this… cloudy 0. 4 windy 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? menu P(C) + P(W) - P(W and C) = 0. 4 + 0. 3 - 0. 1 =0. 6 That middle 0. 1 is being counted twice. So we must subtract it

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 menu B

PROBABILITY BASICS WHAT IF THEY ARE MUTUALLY ECLUSIVE? 0. 4 0. 3 For any 2 events A and B, that are mutually exclusive P(A or B) = P(A) + P(B) 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

PROBABILITY BASICS CARDS: MISC. stuff you have to know in probability. DICE: there are 52 in a deck A regular die has 6 sides. there are 4 suits: (hearts (red), diamonds (red), spades (black), clubs (black)) Each side has a number (1 -6) 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. menu each side has an equal chance of being rolled.

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? How many total possible outcomes there Any face card? are. (there are 52 possible different draws) Not a diamond? menu

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

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

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? menu All draws fall into one of these two categories, so what should their probabilities add to?

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

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: Second flip: H H, T T, H T, T menu T H First flip: T H T

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: Second flip: H H, T T, H T, T menu T H First flip: T H T

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: Second flip: H H, T T, H T, T menu T H First flip: T H T

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: Second flip: H H, T T, H T, T menu T H First flip: T H T

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

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 CONDITIONAL The first thing does NOT affect the second INDEPENDANT menu

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. T H H The probability of getting “H” then “T” is menu T

EVENTS OCCURING TOGETHER The But first the probability affects the of second: the second draw would be much CONDITIONAL different if. . AI asked jar is filled with 5 red, andone 5 green Beans. you to draw a RED then Jelly a GREEN one. What are the chances of randomly pulling out a green one? 0. 555 five green total So the first draw affected the eaten odds in the second After you have pulled out and that green one, draw. THAT is the what conditional probability means what are chances of randomly pulling out a second green one? four green nine total menu

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 menu Assuming this already HAS happened

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? The second JB is GREEN P(G|R) If 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

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

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? 1 or 100% It already happened. menu

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

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

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

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

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

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

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

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