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 V3 V1 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 30 29 28 27 * * * 657,720 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

30P4 Linear Permutations Alternative 1 2 3 4 30 29 28 27 * * * 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

1.5511 x 1025 Linear Permutations 25 24 23 22 21 * * * * . . . 25! 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 1.5511 x 1025 Linear Permutations Alternative 25 24 23 22 21 * * 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 Now consider the circular issue Treat all permutations as if linear C D E A B B C D E A

More Permutations A B C D E F G H I 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 NOT CIRCULAR 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 What is the problem? Are there any others? ABAA BAAA

If all were different, how may ways could we arrange 20 items? 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!

If all were different, how may ways could we arrange 17 items? 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!

! ? ? ? Permutations ORDER Multiply the possibilities Assume the items are in a straight line ! or Use the nPr formula (if no replacement) Are the items in a circle? ? Divide by the number of items in the circle Can the item be turned over? ? Divide by 2 Are there duplicate items in your arrangement? ? Divide by the factorial of the number of each duplicated item

33 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)? 26 ∙ 25 10 9 8 letter number 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? 26 ∙ 10 letter number 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 nCr formula

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

OR  ADD The OR factor. have want want OR 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

OR  ADD The OR factor. have want want OR 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 have want want 5 red 8 blue 4 red OR 4 blue

Probability – “or” { Pr(3R or 3B) = = 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? # of success 5C3 + 8C3 Pr(3R or 3B) = = total # of outcomes 13C3 have want want 5 red 8 blue 3 red { OR 3 blue Total: 13  3

AND  MULTIPLY 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 3 or 4 or 5 blue 3B2NB or 4B1NB or 5B 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 0R5Nr or 1R4NR

PROBABILITY Definition: The ratio  total number of outcomes 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) = = total number of outcomes 2

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) = = = total number of outcomes 6 3

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) = = = total number of outcomes 6 3 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? number of success 5C3 Pr(3R) = = total number of outcomes 13C3 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? number of success 8C3 Pr(3B) = = total number of outcomes 13C3 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? number of success 5C1 ● 8C2 Pr(1R2B) = = total number of outcomes 13C3 have want 5 red 8 blue 1 red 2 blue Total: 13  3

A jar contains 5 red, 8 blue and 7 white marbles 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 # of success 5C1●8C1●7C1 Pr(1R,1B,1W) = = total # of outcomes 20C3 have want 5 red 8 blue 7 white and 1 red 1 blue and 1 white Total: 20  3

A jar contains 7 red, 5 blue and 3 white marbles 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? # of success 7C2 ● 3C2 Pr(2R,2W) = = total # of outcomes 15C4 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 Five cards are dealt from a standard deck of cards. What is the probability that 3 hearts and 2 clubs are obtained? # of success 13C3 ● 13C2 Pr(3H,2C) = = total # of outcomes 52C5 have want 13 diamonds 13 hearts 13 clubs 13 spades and 3 hearts 2 clubs Total: 52  5

Probability – “or” { Pr(3R or 3B) = = 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? # of success 5C3 + 8C3 Pr(3R or 3B) = = total # of outcomes 13C3 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 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) = = total # of outcomes 20C3 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) # of success 26C2 + 4C2 – 2C2 = total # of outcomes 52C2 have want want { overlap 26 red 26 black 2 red OR 2 red kings 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 or 3 red 2 red and 1 blue 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(1R2B or 2R1B or 3R) = Pr(at least 1Red) = 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

Probability – “at least” A jar contains 5 red and 8 blue marbles. If 3 marbles are selected at random, what is the probability that NO red marbles are selected? 8C3 Pr(0R3B) = 13C3 In the previous example we found have want 5 red 8 blue 3 blue Pr(success) + Pr(failure) = 1 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? Pr(success) + Pr(failure) = 1 Pr(success) = 1 - Pr(failure) Pr(>1 red) = 1 – Pr( 0 red ) Pr(3 blue)

Probability – “at least” A jar contains 8 red and 9 blue marbles. If 7 marbles are selected at random, what is the probability that at least one red marbles is selected? FASTEST success Pr(at least 1Red) Pr(1R6B or 2R5B or 3R4B or 4R3B or 5R2B or 6R1B or 7R) failure Pr(0Red) Pr( 0R7B ) Pr(at least 1Red) = 1 - Pr(0R7B) =

Probability – “at least” A jar contains 8 red, 9 blue and 3 white marbles. If 7 marbles are selected at random, what is the probability that at least three red marbles are selected? FASTEST success Pr(> 3Red)  Pr(3-7 red) failure Pr(< 3Red)  Pr(0-2 red) 1 - Pr(0R7NR or 1R6NR or 2R5NR)

Probability – “with replacement” A jar contains 5 red and 8 blue marbles. If 3 marbles are selected at random, what is the probability that one red followed by two blue marbles are selected if each marble is replaced after each selection? Note: In this example an order is specified Must use fractions! R B B

Probability – “with replacement” A jar contains 5 red and 8 blue marbles. If 3 marbles are selected at random, what is the probability that one red and two blue marbles are selected if each marble is replaced after each selection? Problem: Fractions imply order! Must use fractions! R B B Must account of any order!