Introduction to Probability Honors Geometry Summer School

Apply Counting Principle and Permutations Permutation: the number of arrangements of n objects taken r at a time – Order matters, ie 123 is different from 321 Factorial: repeated multiplication of integers from n to 1. – n!=(n)(n-1)(n-2)…(2)(1)

Fundamental Counting Principle Two Events: If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is mn. Three of More Events: the FCP can be extended to three or more events. For example, if three events occur in m, n, and p ways, then the number of ways that all three events can occur is mnp.

Example 1 You are buying a pizza. You have a choice of 3 crusts, 4 cheeses 5 meat toppings, and 8 vegetable toppings. How many different pizzas with one crust, one cheese, one meat, and one vegetable can you choose? “Events” Crust Cheese Meat Vegetable “Occur” 3 x 4 x 5 x 8 “Total” 480

Example 2 A town has telephone numbers that all begin with 646 followed by four digits. How many different phone numbers are possible (a) if numbers can be repeated and (b) if numbers cannot be repeated? (a)10 x 10 x 10 x 10 = 10,000 (b)10 x 9 x 8 x 7 = 5040

Example 3 Eight teams are competing in a baseball playoff. (a)In how many different ways can the baseball teams finish the competition? (b)In how many different ways can 3 of the baseball teams finish first, second, and third? 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 8! = 40,320 8 x 7 x 6 = 336

Permutations of n objects taken r at a time The number of permutations of r objects taken from a group of n distinct objects is denoted by n P r

Example 4 You have 6 homework assignments to complete over the weekend. However, you only have time to complete 4 of them on Saturday. In how many orders can you complete 4 of the assignments? 6 P 4 = 360

Permutations with Repetition The number of distinguishable permutations of n objects where one object is repeated s 1 times, another is repeated s 2 times, and so on is: n!. (s 1 )!(s 2 )!

Example 5 Find the number of distinguishable permutations of the letters in (a) EVEN and (b) PENNSYLVANIA. (a)EVEN 4!. 2! (b) PENNSYLVANIA 12!. (3!)(2!) =12 =39,916,800

Use Combinations Combination: the number of groups that can be taken from n objects, r at a time – Order is not important, 123 is the same as 321 Combinations of n objects taken r at a time – The number of combinations of r objects taken from a group of n distinct objects is denoted by n C r

Example 1 You are picking 7 books from a stack of 32. If the order of the books you choose is not important, how many different 7 book groups are possible? 32 C 7 = 3,365,856

Example 2 The local movie rental store is having a special on new releases. The new releases consist of 12 comedies, 8 action, 7 drama, 5 suspense, and 9 family. (a)You want exactly 2 comedies and 3 family movies. How many different movie combinations can you rent? (b)You can afford at most 2 movies. How many movie combinations can you rent? ( 12 C 2 ) x ( 9 C 3 ) = 66 x 84 = 5544 ( 41 C 0 ) + ( 41 C 1 ) + ( 41 C 2 ) = = 862

Example 3 A popular magazine has 11 articles. You want to read at least 2 of the articles. How many different combinations of articles can you read? We will skip this example.

Define and Use Probability Probability – The likelihood of an event occuring Theoretical Probability – Number of ways a particular event can occur divided by the number of ways any event can occur Odds – Number of ways a particular event can occur divided by the number of ways the event cannot occur Experimental Probability – The number of times an event occurs divided by the total number of attempts Geometric Probability – We will cover this topic later in the course

Example 1 You roll a standard six-sided die. Find the probability of (a) rolling a 5 and (b) rolling an even number. (a) P(5)= (b) P(even)= 1/6 = = 16.7% 3/6 = 0.5 = 50%

Example 2 A cereal company plans to put 5 new cereals on the market: a wheat cereal, a rice cereal, a corn cereal, an oat cereal, and a multigrain. The order in which the cereals are introduced will be randomly selected. Each cereal will have a different price. (a)What is the probability that the cereals are introduced in order of their suggested retail price? (b)What is the probability that the first cereal introduced will be the multigrain cereal? 1/ (5P5)= 1/120 1/5

Odds in favor or odds against an event When all outcomes are equally likely, the odds in favor of an event A and the odds against an event A are: – Odds in favor of event A = – Odds against event A = Number of ways A can occur. Number of ways A cannot occur Number of ways A can occur

Example 3 A marble is drawn from a bag containing 6 red, 12 yellow, and 9 black marbles. Find (a) the odds in favor of drawing a red marble (b) the odds against drawing a black marble. Number of red = 6 = 2 Number of nonred 217 Number of nonblack = 18 = 2 Number of black 9 1

Experimental Probability When an experiment is performed that consists of a certain number of trials, the experimental probability of an event A is given by: Number of time A occurs Number of total trials

Example 4 Exam grades of students in a history class are shown in the bar graph. Find the probability that a randomly chosen student in this history class received a C or better =

Find Probabilities of Disjoint and Overlapping Events Compound Event: the union or intersection of two events Disjoint or mutually exclusive events: events that have no outcomes in common

Probability of Compound Events If A and B are two events, then the probability of A or B is: P(A) + P(B) – P(A and B) Is A and B are disjoint events, then the probability of A or B is: P(A) + P(B)

Example 1 You roll a six-sided die. What is the probability of rolling a 2 or a 5? P(2) + P(5) – P(2 and 5) = 1/6 + 1/6 – 0/6 = 1/3

Example 2 You roll a six-sided die. What is the probability of rolling an odd number or a number less than 3? P(odd) + P(< 3) – P(odd and < 3) = 3/6 + 2/6 – 1/6 = 2/3

Example 3 In a survey of 300 students, 150 like pop music or country music. There are 97 students who like pop music and there are 83 students who like country music. What is the probability that a randomly selected student likes both pop and country music? P(pop or country) = P(pop) + P(country) – P(pop and country) 150/300 = 97/ /300 – P(pop and country) -30/300 = -P(pop and country) 1/10 = P(pop and country)

Example 4 When two six-sided dice are rolled, there are 36 possible outcomes. Find the probability that the sum is not 4 and the sum is greater than or equal to 3. P(sum not 4 and > 3) = 32/36 = 8/

Probability of the Complement of an Event The probability of the complement of A is: P(not A) = 1 – P(A) also denoted P(A)

Example 5 You roll a six-sided die, what is the probability that you do not roll a prime number? P(not prime) = 1 – P(prime) = 1 – 3/6 = 1/2

Find Probabilities of Independent and Dependent Events Independent Events: events in which the occurrence of one has no effect on the occurrence of the other Dependent Events: events in which the occurrence of one does effect the occurrence of the other Conditional Probability: the probability that B will occur given that A has occurred, denoted P(B|A)

Probability of Independent Events If A and B are independent events, then the probability that both A and B occur is: P(A and B) = P(A) x P(B) this “and” is more of a successive and as opposed to a simultaneous and. More generally, the probability that n independent events occur is the product of the n probabilities of the individual events.

Example 1 Every morning, one student in a class of 24 students is randomly chosen to take attendance. What is the probability that the same student will be chosen three days in a row? 1/24 x 1/24 x 1/24 = 1/13824

Example 2 A manufacturer has found that 2 out of every 500 coffee pots produced are defective. What is the probability that at least one coffee pot is defective in the first 300 pots made? We will skip this example.

Probability of Dependent Events If A and B are dependent events, then the probability that both A and B occur is: P(A and B) = P(A) x P(B given A has occurred) also denoted P(A) x P(B|A)

Example 3 Find the probability that (a) a listed person has blue eyes and (b) a male has blue eyes. Green eyes Blue eyesBrown eyes Hazel eyes Male Female (a) P(blue eyes) = 44 / 200 = 11/50 (b) P(male has blue eyes) = 35 / 100 = 7/20

Example 4 You randomly select two marbles from a bag containing 15 yellow, 10 red, and 12 blue marbles. What is the probability that the first marble is yellow and the second marble is not yellow if (a) you replace the first marble before selecting the second, and (b) you do not replace the first marble? (a) P(yellow then not yellow) = 15/37 x 22/37 = 330/1369 (b) P(yellow then not yellow) = 15/37 x 22/36 = 55/222

Example 5 Your teacher passes around a box with 10 red pencils, 8 pink pencils, and 13 green pencils. If you and two people in your group are the first to randomly select a pencil, what is the probability that all three of you select pink pencils? P(pink, pink, pink) = 8/31 x 7/30 x 6/29 = 56/4495