Fundamental Principles of Probability (7-1) Sample Space: Event: If a sample space has n possible outcomes, all of which are equally likely, then the experiment is called ___________ or _________________. Probability of an event, E: P(E) = Properties: 1) 2) 3) Examples: 1) Suppose a coin is tossed and a die is rolled. a. Determine the sample space. b. List the outcomes of the event “an even number is rolled on the die”. c. Find P(a number more than 4 appears on the die). 2) Suppose 2 fair dice are rolled, find: a. P(doubles) b. P(sum = 12) and list all outcomes. c. P( sum = 13) d. P(sum < 13) e. What sum has the highest probability of occurring? What is the probability? 3) A family has 3 children. a. Determine the sample space. b. Find P(2 girls) c. P(at least one boy) d. P(all three children the same gender) Reminder: Relative Frequency represents the number of times an outcome has occurred out of the number of total outcomes of an experiment. R(x) = 0 means R(x) = 1 means Odds in favor of an event: Odds against an event: Finding the probability when you know the odds is easy! 7-2 Addition Counting Principles I. If two events A and B are mutually exclusive (or disjoint), it means they cannot happen at the same time. Then the number of ways A and B can happen together is shown by: N (A B) = 0 The Addition Counting Principle (Mutually Exclusive form): If two finite sets A and B are mutually exclusive then N(A B) = N(A) + N(B) Probability: II. P(A B) = P(A) + P(B) II. If two events A and B are mutually inclusive, then the number of ways they can happen together is shown by : N (A B) 0 The Addition Counting Principle: (General Form): If two finite sets A and B are mutually inclusive, then N(A B) = N(A) + N(B) - N (A B) Probability: P(A or B) = P(A B) = P(A) + P(B) – P(A B) Examples: 1) Roll 1 die. Let A: rolling a number 2; a) Find P (A B) 2) Roll 1 die. Let A: rolling a number a) Find P (A B) B: rolling a 5 b) Find P (A 2; B) B: rolling an even number b) Find P (A If A is any event, then P(not A) = 1 – P(A) B) 7-3 Multiplication Counting Principles Quiznos has a new offer! You can choose from 4 types of meat, 3 types of cheese, and 3 types of roll for your sandwich. If you choose a different sandwich each day, how many days will it take you to try their entire menu? Multiplication Counting Principle: Let A and B be any finite sets. The number of ways to choose one element from A and then one element from B = For Example: When choosing an outfit you have 6 shirts and 4 pairs of pants to choose from. How many possible outfits are there? Selections with Replacement: Let S be a set with n elements. There are nk possible arrangements of k elements from S with replacement. For Example: The SAT test consists of 25 multiple choice questions with 4 options each and 30 true/false questions. How many ways are there to answer the questions? Definition: For n, a positive integer, n factorial is the product of the positive integers from 1 to n. In symbols, n! = (n) (n – 1) (n – 2) (n – 3)…(3) (2) (1) Selections without Replacement: Let S be a set with n elements. Then there are n! possible arrangements of the n elements without replacement. For Example: Teachers had to rank their choices for Days of Diversity classes. There were 6 choices left. How many different ways could I have ordered my choices? Examples: 1) A quiz has 10 questions, each of which can be answered “always”, “sometimes”, or “never”. If you guess on each question, a. how many possible ways are there to answer the questions on the test? b. what is the probability of answering all questions correctly? 2) How many license plates are possible with four letters chosen from A to Z (except O and I) followed by two digits from 0 through 9? 3) How many orders of finish are there in a 7-horse race? 4) How many ways can 6 books be arranged on a shelf? Factorials Evaluate: 5) 5! 8) 7! – 5! 10) 8! 4! 12) 12! 10!2! 6) 6! 7) 7! 9) 4! + 2! 11) 10! 8! ATM More Review 7-1 to 7-3 Name ________________________ Period _________ Date _______ Show any equations used, if appropriate. In 1-3, a coin is tossed and a die is rolled. 1. Determine the sample space for the experiment. 2. List the outcomes of the event “ the coin lands tails” : 3. Find P( a number less than 4 appears on the die). In 4 and 5, consider the experiment of rolling two fair dice. 4. Find P(sum of 5). 5. True or false: P(doubles) = P(sum of 7) 6. Find the probability that a family of four children has exactly two boys. In 7 and 8, consider the experiment of flipping three fair coins. 7. Can P(HHH) = 1.1? 8. Find P(TTTT). 9. If the odds against winning a contest are 100,000 to 1, what is the probability of winning the contest? 10. A movie theater gives each ticket holder a card with a hidden message. If for every 20 cards given out, 12 say “Try again”, 5 are for free popcorn, 2 are for a discounted ticket, and 1 is for a free ticket, find the probability of each event occurring. In 11 and 12, a card is drawn from a standard deck of cards. 11. What is the probability of drawing a 10 or a red card? 12. What is the probability of not drawing a 10? 13. True or false: If A and B are mutually exclusive events, and P(B) = .47, then P(A) = .53. 14. True or false: If A and B are complementary events, then P(A or B) <1. 15. A single die is tossed. Are the two events mutually exclusive: tossing a 2 and tossing an even number? 16. Two babies are born and are not twins. Are the two events independent: the first baby being a girl and the second baby being a girl? 17. A number is chosen from {1, 2, 3, 4}. Are the two events complementary: choosing a 1, and choosing a 4 or a prime number? 18. Jamie has 3 black pairs, 2 brown pairs, and one gold pair of socks in one drawer. If she randomly picks a pair of socks from the drawer, what is the probability that she will pick a brown pair? 19. A green die and an orange die are tossed. What is the probability that the green die shows a three and the orange die shows a number greater than three? 20. Suppose a state forms its license plates with three letters followed by four digits from 0 to 9, and that the first letter must be either A, B, or C. How many different license plates are possible? 21. How many code symbols can be formed using 4 out of the 5 letters V, W, X, Y, Z if the letters: a) are not repeated? b) can be repeated? c) are not repeated and the code must begin with Y? d) are not repeated and the code must end with YZ? 22. Evaluate each of the following: a) 7! – 5! b) 4! + 2! c) 8! 4! 23. A funhouse at an amusement park has 13 doors inside. Once a person has gone through a door, the person cannot go through it again. How many ways are there to go through the funhouse? 24. A certain new car is available with a choice of four exterior colors, three interior colors, two types of trim, and three types of engines. How many different sets of colors and engines are available? 25. How many ways are there of answering a test (assuming one answers all items) if the test has 50 multiple-choice items, each with 4 choices? Independent Events (7-4) Suppose you flip a coin twice. The result of the first flip has no bearing on the result of the second flip. This is sometimes described as the coin has no “memory”. Independent Events: Events A and B are independent if and only if P( A B) P( A) P(B) Examples: 1) Suppose a fair coin is tossed 4 times. Let A = getting all heads and B = getting all tails. Are A and B independent or dependent? 2) A red and white die are tossed. Let C = the sum is 7 and D = the red die shows a 5. Are C and D independent? 3) A red and white die are tossed. Let C = the sum is 8 and D = the red die shows a 4. Are C and D independent? Note: In general, selections with replacement are independent events because each selection is the same. And selections without replacement are dependent because each selection depends on what was selected previously since it is not replaced. Permutations (7-5) Permutation Theorem: There are n! permutations (or arrangements) of n different elements. Notation: n Pr means the number of permutations of n objects taken r at a time. Permutations Formulas: n Pr OR = n(n – 1)(n – 2)(n – 3) n Pr = Example: Find ... (n – r +1) n! (n r )! 12 P7 Note: 0! = 1 Examples: 1) In a 4-person relay race, the fastest runner typically is the last person to run for the relay team. If that place is set, how many different orders are there for the other runners? 2) How many different four-letter words can be formed from the word EQUATIONS? (The words do not have to make sense.) 3) Solve: x P6 = 20 ( x P4 ) ATM Notes 7.7 Probability Distributions A probability distribution is a function which maps each value of a random variable onto its probability. (Remember that a function consists of ordered pairs: (x, P(x)) Example: The probability distribution P for the sum of two dice is {(2,1/36) (3, 2/36) (4,3/36) (5,4/36) (6,5/36) (7, 6/36) (8, 5/36) (9, 4/36) (10, 3/36) (11, 2/36) (12, 1/36)} The expected value ( ) for the probability distribution is found by = (xi P(xi)) The relative frequency R(x) of an outcome is found by R(x) = frequency of x number of trials The mean value of the relative frequency distribution is found by R(xi)) m= (xi One way to compare the expected value with the mean value is to find the percent error: percent error = expected value – mean value expected value Sections 7.4 – 7.7 1) Write the sample space for a jar of marbles containing 6 marbles – 4 red marked with the numbers 1 through 4 and 2 black marked with the numbers 1 and 2. Suppose you have a spinner with 3 congruent areas labeled 1,2 and 3 and one regular die. Decide if the two events are independent by finding P(A), P(B) and P(A B): a) A= The spinner shows a 3; B= The sum of the spinner and die is over 4. b) A= The spinner shows 1; B= the die shows a 2 2) Imagine a drawer with 10 socks – 5 red, 3 blue and 2 green. a) What is the probability that you will reach in and grab 2 red socks (one at a time) without replacement? b) What is the probability that you will reach in and grab 2 red socks (one at a time) with replacement? 3) Evaluate 5 P 2. Show work. 4) Place in increasing value: 13 P 10, 12 P 11, 13 P 11, 12 P 10 5) How many permutations of the letters in CLASH are there? 6) The American League West division consists of four teams, the Angels, Athletics, Mariners and Rangers. Assuming the A’s finish in first place, in how many ways can the rest of the teams finish? Consider a family with four children: 7) List all possible outcomes in the sample space, including birth orders. 8) Let x = the number of girls in the family. Assuming boys and girls are equally likely, list the pairs in the probability distribution for this experiment. 9) Make a histogram of the probability distribution. 10) Find the expected value of x.