Set Theory, Permutations, Combinations and Probability Algebra 2 Class # 1 Set Theory, Permutations, Combinations and Probability

Boys: Girls: Total: Ratios: Proportion: Fraction: Rate: Percent: Probability:

Set Theory Let Set S be defined as the set of all integers numbers from 1 to 12, Set A is the set of all Odd numbers in Set S, and Set B the set of all Prime numbers in Set S Set S: Set A: Set B: 𝑆𝑒𝑡 𝐴∪𝐵: 𝑆𝑒𝑡 𝐴∩𝐵: 𝑆𝑒𝑡 ~𝐴: 𝑆𝑒𝑡 ~𝐵: 𝑆𝑒𝑡 ~𝐴∩𝐵: 𝑆𝑒𝑡 ~𝐴∪𝐵:

𝑃 𝐴 : 𝑃 𝐵 : 𝑃 𝐴∩𝐵 : 𝑃(𝐴∪𝐵) 𝐼𝑠 𝑃 𝐴∪𝐵 =𝑃 𝐴 +𝑃 𝐵 ?

𝑃 ~𝐴 : 𝑃 ~𝐵 : 𝑃 ~𝐴∩𝐵 : 𝑃 ~𝐴∪𝐵 :

A car dealer sale cars that are either American or hybrid A car dealer sale cars that are either American or hybrid. Some cars are American but not hybrid; some cars are hybrid but not American; and some cars are both American and hybrid at the same time. If there are 10 American cars, 6 hybrids and a total of 13 cars in the warehouse. How many cars are American and hybrid at the same time? If a car is chosen at random, What is the probability that the car is American but not hybrid?

Permutations In how many different ways can 3 persons be seated in 3 chairs? In how many different ways can 5 persons be seated in 3 chairs? How many different words can be written with the letters of the word FLORIDA ?

The chart below is to be painted using three different colors The chart below is to be painted using three different colors. If there are seven colors available, in how many different ways cant the chart be painted? Color 2 2 3 1

Formula for permutations 𝒏𝑷𝒓= 𝒏! 𝒏−𝒓 ! 𝟓! 𝟔! 𝟑! 5𝑃3: 7𝑃4 20𝑃10 5𝑃5

Using Permutations to Calculate Probability Every student in your school is assigned a four-digit code, such as 6953, to access the computer system. In each code, no digit is repeated. What is the probability that you are assigned a code with digits 1,2,3, and 4 in any order?

A certain motorcycle license plate consists of 5 digits that are randomly selected. No digit is repeated. What is the probability of getting a license plate consisting of all even digits? There are 8 finalists in the 100-meter dash at the Olympic Games. Suppose 3 of the finalist are from the United States, and that all finalists are equally likely to win. 1- What is the probability that the US will win all 3 medals in this event? 2- What is the probability that the US will win no medal in this event?

Probability with permutations with repetitions The school jazz band has 4 boys and 4 girls, and they are randomly lined up for a yearbook photo 1- Find the probability of getting an alternating boy-girl arrangement 2- Find the probability of getting all the boys grouped together 2/8P8/4!4! 5/8P8/4!4!

Combinations Permutation: In how many different ways can you make a committee of 3 student (a president a secretary and a treasurer) out of a class of 10 students? Combination: In how many different ways can you make a committee of 3 students out of a class of 10 students?

A restaurant offers 8 side dishes (beets, potatoes, carrots, salad, rice, broccoli, cole slaw , and apple sauce). When you order an entrée, you can choose 3 of the side dishes. In how many ways can you choose 3 side dishes? Suppose the restaurant offers a special on Mondays that allows you to choose 4 side dishes. In how many ways can you choose the side dishes?

Formula for Combinations: 𝒏𝑷𝒓= 𝒏! 𝒓! 𝒏−𝒓 ! 5𝐶3: 7𝐶4 20𝐶10 5𝐶5

Combinations and Probability There are 5 boys and 6 girls in a school play. The director randomly chooses 3 of the students to meet with a costume designer. What is the probability that the director chooses all boys? 5𝐶3/11𝐶3

Five cards are randomly drawn from a standard deck of cards Five cards are randomly drawn from a standard deck of cards. What is the probability that all five cards are diamond? 13C5/52C5

Fundamental Counting Principle In how many ways Mr. Smith can be dressed, if there are 5 shirts, 3 pants, 4 pair of shoes and 3 hats in his wardrobe

In Puerto Rico a car license plate is made of 3 letters and 3 numbers In Puerto Rico a car license plate is made of 3 letters and 3 numbers. How many different plates can be issued if: 1- The letters and the digits can be repeated: 2- The letters can not be repeated but the digits no: 3- Neither the letters nor the numbers can be repeated:

A travel agent is offering a vacation package A travel agent is offering a vacation package. Participants choose a type of tour, a meal plan, and a hotel class from the table below. How many different vacations plans can be offered? Tour Meal Hotel Walking Restaurant 4-Star Boat Picnic 3-Star Bicycle 2-Star 1-Star

Using Fundamental Counting Principle to find S A coin is tossed 4 times. 1- What is the probability of getting exactly 3 heads? 2- What is the probability of getting at least 3 heads? 3- What is the probability of getting at most 3 heads? S= 2^4 (4C3)/(16) (4C3+4C4)/16 (4C0+4C1+4C3+)/16

Probability experiments involving two dices (complement of an event)

Probability experiments involving deck of cards

Probability experiments involving spinners

Probability experiments involving bags of marbles

Probability and word problems (complement of an event) There are Blue, Red and Green marbles in a jar. The probability of choosing a green marble at random is 1/3, the probability of choosing a Blue marble is 1/5. What is the probability of choosing a Red marble?

There is an Aquarium containing snails, fishes, and shrimps There is an Aquarium containing snails, fishes, and shrimps. If the probability of choosing a crustacean is 1/3, of choosing a mollusk is 2/7. What is the probability of choosing an animal that is not an invertebrate?

Conditional Probability 𝑷 𝑨 𝑩 = 𝑷(𝑨∩𝑩) 𝑷(𝑩) 𝑷 𝑨 𝑩 = 𝑷(𝑨∩𝑩) 𝑷(𝑩) Took Medicine Not Medicine Total Not Cured 5 22 27 Cured 60 13 73 65 35 100 P(c)= P (nc)= P(m)= P(c ∩ m)= P(nc ∩ m)= P(c/m)= P(nc/m)=

To the nearest percent what is the probability that a student who failed the exam got less than 6 hours sleep? What is the probability that a student who got less than 6 hours of sleep failed the exam? Passed exam Failed Total Less than 6 hours sleep 12 10 22 More than 6 hours sleep 90 8 98 102 18 120

For a standard deck of playing cards, find the probability that a red card randomly drawn from the deck is a jack?

Independent and dependent events Suppose you pick one marble at random what is the probability that the marble is blue P(b)? and what is the probability that the marble is red? P(r) If you put the marble back in the bag and pick a second marble what is the probability that the second marble is red given that the first was blue P(r/b)? If you don’t return the first marble to the bag, what is the probability that the second marble is red given that the first was blue P(r/b)? Independent even when P(B/A)_____P(B) 𝑃 𝐴∩𝐵 =𝑃(𝐴)∙𝑃(𝐵) Dependent events is when P(B/A)_____P(B) 𝑃 𝐴∩𝐵 =𝑃 𝐴 .𝑃 𝐵 𝐴

Examples If you pick two marbles one after the other with replacement: 1- What is the probability that the first is Blue and the Second is Red? 2- What is the probability that both the first and the second are Blue? If you pick two marbles one after the other without replacement? 3- What is the probability that the first is Blue and the second Red? 4- What is the probability that both the first and the second are Blue?

Skydiver Landing probability P(A)= P(B)= P(𝐴∩𝐵)= P(𝐴∪𝐵)= P(A/B)= P(B/A)= A B

Skydiver Landing probability P(A)= P(B)= P(𝐴∩𝐵)= P(𝐴∪𝐵)= P(A/B)= P(B/A)= B

Skydiver Landing probability P(A)= P(B)= P(𝐴∩𝐵)= P(𝐴∪𝐵)= P(A/B)= P(B/A)= B A

The Bayes’ Theorem

What is the probability that a person that took the medicine was not cured Took Medicine Not Medicine Total Not Cured 5 22 27 Cured 60 13 73 65 35 100