COMBINATORICS AND PROBABILITY Chapter 13

Section 13.1 Permutations and Combinations  Distinguish between Dependent and Independent Events  Define and understand permutations and combinations

13.1 Coin Activity  How many different ways can you arrange three jellybeans

Section 13.1 Tree Diagram  A way to show all possible choices Independent Event  Events that do not effect each other  Examples: Math grade and English Grade Dependent Event  Events that do effect each other  Semester grades and Final Grade

Section 13.1 Basic Counting Principle  INDEPENDENT events  P Different Ways for event 1  Q Different Ways for event 2  P*Q Different ways of both choices

Section 13.1 Mrs. Innerst needs to choose what to wear to WOHS’s Prom. She has the choice of 3 different dresses, 6 different pairs of shoes, and 10 different hairstyles. She also needs to pick one of 7 different purses and one of 20 different dates.  Are these events dependent or independent?  Independent  How many different selections are available for Mrs. Innerst?  20*7*10*6*3

Section 13.1 Permutations  The arrangement of objects in a certain order  Order of objects is very important!  The number of permutations of n objects taken n at a time  P(n,n)=n!  The number of permutations of n objects taken r at a time  P(n,r)= n! (n-r)!

Section 13.1 You want to rank your four semester teachers in order from favorite to least favorite.  Is this list independent or dependent?  Dependent  How many ways can you list them in different orders?  P(4,4)  4!

Section 13.1 There are 100 kids in the junior class and it is time to vote for class officers. There are positions for President, Vice President, Secretary, and Treasurer. How many ways can these positions be elected?  P(100,4)  100!/(100-4)!  96!  (overflow in calculator)

Section 13.1 Combination  The order of selections/events DOES NOT matter  C(n,r)= n! (n-r)!*r!

Section 13.1 Difference between Combination and Permutation  For Permutation  ORDER MATTERS  For Combination  ORDER DOES NOT MATTER  Think of combining ingredients for cake, order doesn’t matter

Section 13.1 I want to select five students from the class, with a total of 20 students, to do problems on the board. How many different groups of students can I pick?  Does order matter?  No  Combination or Permutation? Combination  C(20,5)  20!/(15!*5!)  15, 504 different groups of 5 students

Section 13.1 The math club has 20 members of which 9 are male and 11 are female. Seven members will be selected to go to a math competition. How many teams of 4 females and 3 males can be formed?  Is order important?  No, Combinations  C(9,3)*C(11,4)  27, 720 different team possibilities  How many ways can a president and vice president be chosen for the team?  Does order matter? Yes, Permutation P(7,2) 42 ways

Section 13.1 How many ways can a coach select a starting team of one center, two forwards, and two guards if the basketball team consists of three centers, five forwards, and three guards?  C(3,1)*C(5,2)*C(3,2)  90 Different Ways

Section 13.1 Calculator buttons  !  nPr  nCr

Section 13.2 Permutations with Repetitions and Circular Permutations

Section 13.2 Permutations with Repetitions  N objects, P are alike and Q are alike  N! P!*Q!

Section 13.2 How many ways can you arrange 8 jellybeans by color where 3 are pink?  How many ways can they be arranged normally?  P(8,8)=8!  How many are the same? 33  8!/3!, or 6720 different ways

Section 13.2 How many twelve letter patterns can be formed from the letters of the word cosmopolitan?  12!/3!  79, 883, 600

Section 13.2 Circular Permutations  When objects are arranged in a circle with no reference point  N! or (n-1)! N

Section 13.2 There are 8 pieces of pizza each with different toppings in a large pie. How many ways can the pieces be arranged?  Is this circular or linear?  circular  (8-1)!  7!  5,040 different ways There are 13 kids seated on a merry-go-round. How many different ways can the kids be arranged?  (13-1)!  12!

Section 13.2 What if their was an attendant who wanted to collect tickets and he started with a brown horse, assuming all horses are different colors. How many possible arrangements are their relative to the brown horse?  Is this linear or circular?  Linear since we now have a point of reference  13!

Section 13.3 Probability and Odds

Section 13.3 Probability  The measure of the changes of an event happening Sample Space  The set of all outcomes Success  The desired outcome Failure  Any other outcome rather than the desired outcome

Section 13.3 Probability of Success and Failure  P(S) = S s+f  P(F) = F s+f  P(S) + P(F) = 1

Section 13.3 A deck of cards has 52 cards total. What is the probability of pulling a heart?  How many hearts?  13  Probability of pulling a heart?  13/52  What would be the probability of not pulling a king?  48/52

Section 13.3 A class contains 8 boys and 7 girls  What is the probability that a girl is called on for a question?  7/15 What is the probability that you role a 6 on a die?  1/6

Section 13.3 There are 10 IPods in a basket. 3 of those IPods don’t work. If you selected 3 IPods at random, what is the probability that all three are defective?  P(3 defective ipods)=Ways of selecting 3 defective Ways of selecting 3 ipods  P(3 defective ipods)= C(3,3) C(10,3)  P(3 defective ipods)= 1 120

Section 13.3  In an AFM Class there are 40 students total, 13 of which are currently failing. If 5 students are chosen at random, what is the probability that at least 1 is failing?  P(at least 1 failing student)=1-P(no failing students)  P(no failing students) = C(27,5) C(40,5)  P(no failing students) =   P(at least 1 failing student)= 1-P(no failing students)  P(at least 1 failing student)= The probability of picking at least one failing student is 87.7%.

Review 13.3/13.2 A box contains 3 tennis balls, 7 softballs, and 11 baseballs. One ball is chosen at random.  What is the probability that it is not a baseball?  10/21 Of 7 kittens in a litter, 4 have tiger stripes. Three kittens are picked at random. Find the probability of choosing only ONE kitten with stripes.  C(4,1)*C(3,2) C(7,3)  12/35

Review 13.3/13.2 Of 7 kittens in a litter, 4 have tiger stripes. Three kittens are picked at random. Find the probability of choosing all three that have stripes.  C(4,3) C(7,3)  4/35 How many different ways can the letters of the word Kangaroo be arranged?  8!/(2!*2!)  10,080 Determine the number of arrangements of 11 football players in a huddle  (11-1)!  3,628,800 ways because circular

Section 13.3 Odds  The odds of a successful outcome of an event is the ratio of the probability of its success to the probability of its failure.  Odds = P(S) P(F)

Section 13.3 Twelve male and 16 female students have been selected as candidates for college scholarships. If the awarded recipients are to be chosen at random, what are the odds that 3 will be male and 3 will be female?  Total number of possible groups  C(12,3)  220  C(16,3)  560  How do we find the total possible number of groups?  220*560=123, 200 possible groups

Section 13.3  Total number of qualifiers?  C(28,6)  376,740 groups of 6 who qualified Number of groups who qualified that were not 3 male and 3 female?  376, ,200=253,540  Odds=P(S)/P(F)  P(S)=123,200/376,740   P(F)=253,540/376,740   Odds=0.327/0.673   close to 1:2.

Section Kittens in a litter and only 4 have stripes. What is the odds of picking one that is not striped.  P(S)=C(4,2)*C(3,1) C(7,3)  P(S)=18/35  P(F)=1-P(S)=17/35  Odds: (18/35)/(17/35)  Odds: 1:4

Section 13.3 Of 27 students in a class, 11 have blue eyes, 13 have brown eyes, and 3 have green eyes. If 3 students are chosen at random what are the odds of 2 having brown eyes and 1 having blue eyes?  P(S)=C(13,2)*C(11,1) C(27,3)  P(S)=858/2925  22/75  P(F)=1-22/75=53/75  Odds:(22/75)/(53/75)  22/53

Section 13.4 Probabilities of Compound Events

Section 13.4 Probability of two independent events A and B.  P(A and B)=P(A)*P(B)

Section 13.4 Using a standard deck of playing cards, find the probability of drawing a king, replacing it, then drawing a second king.  Are these independent events?  Yes, because card is replaced  P(A and B) =P(A)*P(B)  P(A) = 4/52  1/13  P(B) = 4/52  1/13  P(A and B) = 1/13*1/13=1/169

Section 13.4 Find the probability of rolling a sum of 7 on the first toss of two dice an a sum of 4 on the second toss.  Are these independent events?  P(7)=6/36  P(4)=3/36  P(7 and 4)=18/1296  1/72

Section 13.4 Probability of Two Dependent Events A and B  P(A and B)=P(A)*P(B following A)

Section 13.4 What is the probability of randomly selecting two navy socks from a drawer that contains 6 black and 4 navy socks?  Dependent because not replacing  P(A)=4/10*3/9

Section 13.4 Probability of Two Mutually Exclusive Events  Mutually Exclusive means?  Two events that can not happen at the same time  Venn Diagram Example  P(A or B) = P(A) + P(B)

Section 13.4 You are a contestant in a game where if you select a blue ball or red ball you get a million dollars. You must select the ball at random from a box containing 2 blue, 3 red, 9 yellow, and 10 green balls. What is the probability that you will win the money?  Mutually exclusive since you can not pick two balls at once  P(Blue or red)=P(blue)+P(red)  P(Blue)=2/24  P(Red)=3/24  P(Blue or red)= 5/24

Section 13.4 Probability of Inclusive Events  Inclusive means?  Events that are not mutually exclusive and can overlap  Venn Diagram Example  P(A or B) = P(A) + P(B) – P(A and B)

Section 13.4 The probability for a student to pass the road test for their license the first time is 5/6. The probability of passing the written part on the first attempt is 9/10. The probability of passing both the road and written tests on the first attempt is 4/5.  Are these events mutually exclusive or mutually inclusive?  Mutually Inclusive since it is possible to pass both events

Section 13.4 What is the probability that you can pass either part on the first attempt?  P(Passing road)=5/6  P(Passing written)=9/10  P(Passing both)=4/5  P(Passing either)=5/6+9/10-4/5  P(Passing either)=14/15

Section 13.4 There are 5 students and 4 teachers on a committee. A group of 5 members is being selected to attend a workshop. What is the probability that the group attending the workshop will have at least 3 students?  Mutually Exclusive because group of 3 students OR 4 students OR 5 students.  P(At least 3 students) = P(3 students) + P(4 students) + P(5 students)  C(5,3)*C(4,2) + C(5,4)*C(4,1) + C(5,5)*C(4,0) C(9,5) C(9,5) C(9,5) =60/ / /126 =9/14

Conditional Probability SECTION 13.5

Conditional Probability The probability of an event under the condition that some preceding event has occurred  P(A l B)=P(A and B) P(B)

Conditional Probability You toss two coins. What is the probability that you toss two heads given that you have tossed at least 1 head?  P(A) = The two coins come up heads  P(B) = There is a least one head  Different outcomes of two coins?  (H,H)-(T,H)-(H,T)-(T,T)  P(B)=3/4  P(A and B) = ¼  P(A l B) = P(A and B)/P(B)  P(A l B) = (¼)/(3/4)  P(A l B) = 1/3

Conditional Probability A neighborhood lets families have two pets. They can have two dogs, two cats, or one of each. What is the probability that the family will have exactly 2 cats if the second pet is a cat? P(A) = Two cats = 1/3 P(B) = At least one cat=2/3 P (A l B) = (1/3)/(2/3)=1/2

Conditional Probability Two number cubes are tossed. Find the probability that the numbers showing on the cubes match given that their sum is greater than five. P(A) = Cubes Match  P(A) = 1/6 P(B) = Sum is greater than 5  P(B)=26/36 P(A and B) = 4/36 P(A l B) = (4/36)/(26/36) P(A l B) = 2/13

Conditional Probability One card is drawn from a standard deck of cards. What is the probability that it is a queen given that it is a face card?  P(A) = Queen = 4/52  P(B) =Face card = 12/52  P(A and B) = 4/52  P(A l B) = (4/52)/(12/52)  P(A l B) = 1/3

Section 13.6 The Binomial Theorem and Probability

Section 13.6 Binomial Expansion  (X+Y) 3  X 3 +3x 2 y+3xy 2 +y 3  Coefficients for Exponents following Combinations for  C(3,3)  X 3  C(3,2)  3x 2 y  C(3,1)  3xy 2  C(3,0)  y 3

Section 13.6 Binomial Experiments exists if and only if:  Each trial has exactly two outcomes  There must be a fixed number of trials  The outcomes of each trial MUST BE INDEPENDENT  The probabilities in each trial are the same

Section 13.6 Eight out of every 10 persons who contract an infection can recover. If a group of 7 people become infected what is the probability that exactly 3 people with recover?  Total of 7 People  Two outcomes: R-Recovery, N –Not recovered  P(R) = 80%, P(N)=20%  Events are independent  Binomial  (R+N) 7  What term represents 3 people recovering? R3N4R3N4

Section 13.6 R 3 N 4 C(7,3)*(0.8) 3 *(0.2) 4 35*(0.8) 3 *(0.2) 4 =

Section 13.6 In Lisa’s art class, 1 out of 5 paintings that she makes will be chosen for an art show. If she is preparing 9 paintings for the competition, what is the probability that exactly 2 of them will be chosen?  (P+NP) 9  C(9,2)*P 2 *NP 7  36*(0.2) 2 *(0.8) 7   30.2%

Section 13.6 A weather reporter is forecasting a 30% chance of rain for today and the next four days. What is the probability of not having rain on any day?  (R+D) 5  C(5,0)*R 0* D 5  1*0.3 0 *0.7 5  16.8%

Section 13.6 What is the probability of having rain no more than three of the five days?  (R+D) 5  1-[P(rain on 4 days)+ P(Rain on 5 days)]  1-[(C(5,4)*.3 4 *.7 1 )+(C(5,5)*.3 5 *.7 0 )]  1-[( )]  96.92%