Counting Principle If one activity can occur in m way and another activity can occur in n ways, then the activities together can occur in mn ways. Permutations arrangements of objects in a specific order np r = n! (n r)! Combinations an arrangement of objects in which the order is not important nc r = n! (n r)!r! In how many different orders can the program for a music recital be arranged if 7 students are to perform? In how many ways can one junior and one senior be selected from a group of 8 juniors and 6 seniors? A reading list gives the titles of 20 novels and 12 biographies from which each student must choose 3 novels and 2 biographies to read. How many different ways can these titles be chosen? A choral group is composed of 6 juniors and 8 seniors. If 2 students are chosen at random to sing a duet, what is the probability that a junior and a senior are selected? 1
In my third period class there are 12 boys and 14 girls. Find the number of ways I can select a team of 8 students from the class to work on a group project if the teams must consist of 6 girls and 2 boys. How many different arrangements are possible with the letters in the word senselessness? A bag contains 8 red marbles, 4 green marbles, and 3 black marbles. You select 5 marbles at random. What is the probability that your selection contains: a. only red marbles b. 2 green marbles c. 2 black marbles and 1 green marble A school group consists of 4 first year students, 8 sophomores, 2 juniors and 5 seniors. How many possible subgroups of 7 can I make from this larger group? What is the probability that this subgroup will contain 2 first year students, 3 sophomores, 1 junior and 1 senior? Given the set of numbers {1, 4, 5, 7 8}, if each digit is used only once, how many different a. four digit numbers are possible? b. Four digit odd numbers can be formed? c. Three digit numbers larger than 700 can be formed? d. Even three digit numbers greater than 800 can be formed? 2
Solve for x: x P 7 = 12 x P 5 3
Bernoulli Experiments Probability A multi stage experiment comprised of independent events, each of which must have a definite outcome. FORMULA nc r p r q n r where n = the total number of trials r = the number of times you get what the questions wants i.e. the number of successes p = the probability of you getting what the question asked for i.e. the probability of a success q = the probably of getting what you don t want i.e. the probability of a failure *** NOTE : p + q must equal 1 *** Example: 1. You toss a fair six sided die 4 times. What is the probability of getting a five three times? 2. The probability of a student eating breakfast in the morning is ¾. What is the probability that in 10 randomly selected mornings, she eats breakfast exactly 7 times? 4
Bernoulli Experiments 3. Mrs. Gruber gave her history class a multiple choice quiz containing six questions. A student must answer at least four correct to pass the quiz. Greg decided to guess on every question. If each of the four possible answers to each question is equally likely to be chosen, what is the probability that Greg passed the quiz? 2. a. If a letter is selected at random from the name MARILYN in five separate trials, what is the probability that the M is chosen exactly three times? b. If a letter is selected at random from the name DAPHNE in seven separate trials, what is the probability that a vowel is chosen at least six times? c. If a letter is selected at random from the name NORMA in six separate trials, what is the probability that a consonant is chosen at most once? 5
5. According to a federal agency, when a lie detector test is given to a truthful person, the probability that the test will show that the person is not telling the truth is 20%. If a company interviews five truthful candidates for a job and asks about thefts from prior employers, what is the probability a lie detector test will show that at most one candidate is not telling the truth? 6
A circular target with a radius of 9 inches has a bullseye that has a radius of 3 inches. If five arrows randomly hit the target, what is the probability that at least four hit the bull's-eye? 7
Question #17 from the review packet 8