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## Counting and Probability

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**Counting and Probability**By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: April 16, 2008**Fundamental Counting Principal**How many different meals can be made if 2 main courses, 3 vegetables, and 2 desserts are available? Let’s choose a main course 2 M1 M2 Now choose a vegetable 3 x V1 V2 V3V1 V2 V3 Finally choose A dessert 2 x D1 D2 D1 D2 D1 D2 D1 D2 D1 D2 D1 D2 12 1 2 3 4 5 6 7 8 9 10 11 12**Linear Permutations**A club has 30 members and must select a president, vice president, secretary, and treasurer. How many different sets of officers are possible? 30 29 28 27 * * * president vice-president secretary treasurer 657,720**Linear Permutations**Alternative A club has 30 members and must select a president, vice president, secretary, and treasurer. How many different sets of officers are possible? 1 2 3 4 30 29 28 27 * * * Try with your calculator president vice-president secretary treasurer 657,720 30P4**Permutation Formula**657,720**Linear Permutations**There are 25 students in a classroom with 25 seats in the room, how many different seating charts are possible? 25 24 23 22 21 * * * * . . . seat 1 seat 2 seat 3 seat 4 seat 5 1.5511 x 1025 25!**Linear Permutations**Alternative There are 25 students in a classroom with 25 seats in the room, how many different seating charts are possible? 25 24 23 22 21 * * * * . . . seat 1 seat 2 seat 3 seat 4 seat 5 1.5511 x 1025 25! 25P25**Permutation Formula**1.5511 x 1025**More Permutations**There are 5 people sitting at a round table, how many different seating arrangements are possible? straight line Divide by 5 A B C D E E A B C D D E A B C C D E A B B C D E A**More Permutations**REVIEW There are 5 people sitting at a round table, how many different seating arrangements are possible? straight line When circular, divide by the number of items in the circle A B C D E E A B C D D E A B C Treat all permutations as if linear Now consider the circular issue C D E A B B C D E A**More Permutations**There are 9 people sitting around a campfire, how many different seating arrangements are possible? straight line Yes, divide by 9 Treat all permutations as if linear Is it circular? A B C D E F G H I**More Permutations**There are 5 people sitting at a round table with a captain chair, how many different seating arrangements are possible? straight line NOT CIRCULAR A B C D E E A B C D D E A B C NOTE: C D E A B B C D E A Each table has someone different in the captian chair!**More Permutations**How many ways can you arrange 3 keys on a key ring? straight line Yes, divide by 3 Treat all permutations as if linear Is it circular? A B C Now, try it. . . PROBLEM:Turning it over results in the same outcome. So, we must divide by 2.**More Permutations**How many ways can you arrange the letters MATH ? How many ways can you arrange the letters ABCDEF ?**Permutations with Repetition**How many ways can you arrange the letters AAAB? Divide by 3! Let’s look at the possibilities: If a permutation has repeated items, we divide by the number of ways of arranging the repeated items (as if they were different). AAAB AABA Are there any others? What is the problem? ABAA BAAA**How many ways can you arrange 5 red, 7 blue and 8 white**flags on the tack strip across the front of the classroom? If all were different, how may ways could we arrange 20 items? There are 5 repeated red flags Divide by 5! There are 7 repeated blue flags Divide by 7! There are 8 repeated white flags Divide by 8!**How many ways can you arrange the letters AABBCCCCDEFGGGGGG**? If all were different, how may ways could we arrange 17 items? There are 2 repeated A’s Divide by 2! There are 2 repeated B’s Divide by 2! There are 4 repeated C’s Divide by 4! There are 6 repeated G’s Divide by 6!**!**Assume the items are in a straight line ? Are the items in a circle? ? Can the item be turned over? ? Are there duplicate items in your arrangement? Permutations ORDER Multiply the possibilities or Use the nPrformula (if no replacement) Divide by the number of items in the circle Divide by 2 Divide by the factorial of the number of each duplicated item**How many ways can you put 5 red and 7 brown beads on a**necklace? How may ways could we arrange 12 items in a straight line? Is it circular? Yes divide by 12 Can it be turned over? Yes divide by 2 33 Are there repeated items? Yes divide by 5! and 7!**How many ways can you arrange 5 red and 7 brown beads on a**necklace that has a clasp? How may ways could we arrange 12 items in a straight line? Is it circular? N0 the clasp makes it linear Can it be turned over? Yes divide by 2 396 Are there repeated items? Yes divide by 5! and 7!**How different license plates can have 2 letters followed by**3 digits (no repeats)? A straight line? Is it circular? No 468,000 Can it be turned over? No Are there repeated items? No**How different license plates can have 2 letters followed by**3 digits with repeats? A straight line? Is it circular? No 676,000 Can it be turned over? No Are there repeated items? Yes, but because we are using multiplication and not factorials, we do not need to divide by anything.**Combinations**NO order NO replacement Use the nCrformula**Combinations**An organization has 30 members and must select a committee of 4 people to plan an upcoming function. How many different committees are possible? 27,405**Combinations**A plane contains 12 points, no three of which are co-linear. How many different triangles can be formed? 220**Combinations**An jar contains 20 marbles – 5 red, 6 white and 9 blue. If three are selected at random, how many ways can you select 3 blue marbles? 84**Combinations**An jar contains 20 marbles – 5 red, 6 white and 9 blue. If three are selected at random, how many ways can you select 3 red marbles? 10**The OR factor.**An jar contains 20 marbles – 5 red, 6 white and 9 blue. If three are selected at random, how many ways can you select 3 blue marbles or 3 red marbles? OR ADD**The OR factor.**An jar contains 20 marbles – 5 red, 6 white and 9 blue. If three are selected at random, how many ways can you select 3 red marbles or 3 blue marbles? OR ADD have want want 5 red 6 white 9 blue 3 red OR 3 blue**The OR factor.**An jar contains 13 marbles – 5 red and 8 blue. If four are selected at random, how many ways can you select 4 red marbles or 4 blue marbles? OR ADD want have want 5 red 8 blue 4 red OR 4 blue**Probability – “or”**A jar contains 5 red and 8 blue marbles. If 3 marbles are selected at random, what is the probability that all three are red or all three are blue? 5C3 + 8C3 # of success Pr(3R or 3B) = = 13C3 total # of outcomes have want want 5 red 8 blue 3 red { OR 3 blue Total: 13 3**The AND factor.**An jar contains 20 marbles – 5 red, 6 white and 9 blue. If three are selected at random, how many ways can you select 2 blue marbles and 1 red marble? AND MULTIPLY**At least**An jar contains 20 marbles – 5 red, 6 white and 9 blue. If five marbles are selected at random, how many ways can you select at least 3 blue marbles? 3 or 4 or 5 blue 3B2NB or 4B1NB or 5B**At most**An jar contains 20 marbles – 5 red, 6 white and 9 blue. If five marbles are selected at random, how many ways can you select at most 1 red marbles? 0 or 1 red 0R5Nror 1R4NR**Definition:**PROBABILITY number of success The ratio total number of outcomes**Probability**A coin is tossed, what is the probability that you will obtain a heads? Look at the sample space/possible outcomes: { H , T } number of success 1 Pr(H) = = 2 total number of outcomes**Probability**A die is tossed, what is the probability that you will obtain a number greater than 4? Look at the sample space/possible outcomes: { 1 , 2 , 3 , 4 , 5 , 6 } number of success 2 1 Pr(>4) = = = 6 3 total number of outcomes**Probability – Success & Failure**A die is tossed, what is the probability that you will obtain a number greater than 4? number of success 2 1 Pr(>4) = = = 6 3 total number of outcomes What is the probability that you fail to obtain a number greater than 4? number of failures 4 2 Pr(>4) = = = 6 3 total number of outcomes 1 Pr(success) + Pr(failure) = 1 TOTAL =**Probability**A jar contains 5 red and 8 blue marbles. If 3 marbles are selected at random, what is the probability that all three are red? 5C3 number of success Pr(3R) = = 13C3 total number of outcomes have want 5 red 8 blue 3 red Total: 13 3**Probability**A jar contains 5 red and 8 blue marbles. If 3 marbles are selected at random, what is the probability that all three are blue? 8C3 number of success Pr(3B) = = 13C3 total number of outcomes have want 5 red 8 blue 3 blue Total: 13 3**Probability – “and”**multiply A jar contains 5 red and 8 blue marbles. If 3 marbles are selected at random, what is the probability that one is red and two are blue? 5C1● 8C2 number of success Pr(1R2B) = = 13C3 total number of outcomes have want 5 red 8 blue 1 red 2 blue Total: 13 3**A jar contains 5 red, 8 blue and 7 white marbles. If 3**marbles are selected at random, what is the probability that one of each color is selected? 1 red, 1 blue, & 1 white 5C1●8C1●7C1 # of success Pr(1R,1B,1W) = = 20C3 total # of outcomes have want 5 red 8 blue 7 white and 1 red and 1 blue 1 white Total: 20 3**A jar contains 7 red, 5 blue and 3 white marbles. If 4**marbles are selected at random, what is the probability that 2 red and 2 white marbles are selected? 7C2 ● 3C2 # of success Pr(2R,2W) = = 15C4 total # of outcomes have want 7 red 5 blue 3 white 2 red and 2 white Total: 15 4**Five cards are dealt from a standard deck of cards. What is**the probability that 3 hearts and 2 clubs are obtained? 13C3 ● 13C2 # of success Pr(3H,2C) = = 52C5 total # of outcomes have want 13 diamonds 13 hearts 13 clubs 13 spades and 3 hearts 2 clubs Total: 52 5**Probability – “or”**A jar contains 5 red and 8 blue marbles. If 3 marbles are selected at random, what is the probability that all three are red or all three are blue? 5C3 + 8C3 # of success Pr(3R or 3B) = = 13C3 total # of outcomes have want want 5 red 8 blue 3 red { OR 3 blue Total: 13 3**A jar contains 5 red and 8 blue marbles and 7 yellow**marbles. If 3 marbles are selected at random, what is the probability that all three are the same color? 3 red or 3 blue or 3 yellow ? 5C3 + 8C3 + 7C3 # of success Pr(3R or 3B or 3w) = = 20C3 total # of outcomes have want want want { 5 red 8 blue 7 yellow 3 red OR OR 3 blue 3 yellow Total: 20 3**Probability – “or” with overlap**If two cards are selected from a standard deck of cards, what is the probability that both are red or both are kings? Pr(2R or 2B) = Pr(2R) + Pr(2K) – Pr(2RK) 26C2+ 4C2– 2C2 # of success = 52C2 total # of outcomes have want want { overlap 26 red 26 black 2 red 2 red kings OR 4 kings 48 other 2 kings Total: 52 2**Probability – “and” with “or”**A jar contains 5 red and 8 blue marbles. If 3 marbles are selected at random, what is the probability that two are red and one is blue or that one is red and two are blue? 5C2● 8C1 + 5C1 ● 8C2 # of success Pr(2R1B or 1R2B) = = 13C3 total # of outcomes have want want 5 red 8 blue 2 red 1 red { OR and and 1 blue 2 blue Total: 13 3**Probability – “at least”**A jar contains 5 red and 8 blue marbles. If 3 marbles are selected at random, what is the probability that at least two red marbles are selected? 2 red and 1 blueor 3 red 2 red or 3 red 5C2● 8C1 + 5C3 # of success Pr(2R1B or 3R) = Pr(at least 2Red) = = 13C3 total # of outcomes Wait, we need 3 marbles! have want want 5 red 8 blue 2 red 3 red { OR and 1 blue Total: 13 3**Probability – “at least”**A jar contains 5 red and 8 blue marbles. If 3 marbles are selected at random, what is the probability that at least one red marble is selected? 5C1● 8C2 + 5C2 ● 8C1 + 5C3 Pr(at least 1Red) = Pr(1R2B or 2R1B or 3R) = 13C3 Remember, we need 3 marbles! have want want want { 5 red 8 blue 1 red 2 red 3 red OR OR and and 2 blue 1 blue Total:13 3