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comBINATORICS AND PROBABILITY

comBINATORICS AND PROBABILITY. Combinatorics. Combinatorics is the branch of mathematics studying the enumeration , combination , and permutation of sets of elements and the mathematical relations that characterize their properties. Counting Techniques.

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comBINATORICS AND PROBABILITY

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  1. comBINATORICS AND PROBABILITY

  2. Combinatorics • Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties.

  3. Counting Techniques Fundamental (Multiplication) Principle • Let E1, E2, E3, … , En be a sequence of “n” events. If Ek can occur mk ways for k = 1, 2, 3, … , n, then there are m1•m2•m3• … •mnways for all n events to occur.

  4. Counting Techniques Examples • Given 9 digits 1, 2, 3, 4, 5, 6, 7, 8, 9, how many 3-digit numbers could be formed without repetition of digits? With repetition of digits? • In how many ways can a person enter and exit a building with 5 doors if the person cannot pass through the same door twice? If the person can pass through the same door twice? • How many different ways can two multiple choice questions be answered if there are only two choices/answers in each question?

  5. Factorials (!) Examples: Evaluate the following. • 5! • 6! • 0! • 3! + 4! • 5! 3!

  6. Permutations • Permutation is an arrangement of objects wherein order is taken into account. ORDER MATTERS!

  7. Permutations Permutation of objects taken all at a time nPn = n! Examples • In how many ways can we arrange 3 different books in a shelf? • How many different arrangements of letters can we have from the letters of the word “SURVEY”?

  8. Permutation of n objects taken r at a time nPr = Note: “n” objects to choose from and pick/select “r” of those objects. Examples: Evaluate the following. • 5P3 • 6P6 • 10P4

  9. Permutation of n objects taken r at a time Examples • In how many ways can a coach assign the 5 starting positions in basketball to nine equally qualified men? • Suppose there is a class of 20 students and election are being made for class president and class vice-president. How many different ways could the candidates be picked?

  10. Circular Permutation (one position must be fixed) (n – 1)P(n – 1) = (n – 1)! Example In how many ways can 8 guests be seated in a round table with eight chairs?

  11. Permutations of n objects not all distinct(permutations involving repeated symbols) P = Where N = n1 + n2 +…+nk N = total no. of objects n1, n2, n3, … ,nk = types of objects Example How many different linear arrangements of the letters from the word “MATHEMATICS” are there?

  12. Combinations • Combination is a selection of objects with no attention given to the order of the objects. ORDER DOES NOT MATTER!

  13. Combination of n objects taken all at the same time nCn = 1 Example In how many ways can six members form a committee of six?

  14. Combination of n objects taken r at a time nCr = Examples Evaluate the following. • 5C2 • 8C6 • 3C3

  15. Combination of n objects taken r at a time Examples • In how many ways can 10 individuals be selected from 25 individuals? • In how many ways can a group of 10 people form a committee consisting of five people?

  16. Exercises • Given digits 0, 1, 2, 3, 4, how many 3-digit numbers could be formed without repetition of digits? With repetition of digits? • On a test, a student must select 6 out of 10 questions. In how many ways can this be done? • How many different linear arrangements of the letters from the word MISSISSIPPI are there?

  17. Exercises • Suppose 10 horses run a race. How many different ways could 1st place, 2nd place and 3rd place occur? • Suppose 10 horses run a race. You would like to know in how many ways 3 horses can finish in 1st, 2nd, 3rd in any order. • How many committee of four members can be formed from ten engineers?

  18. Probability Theory • Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. • Probability is the measure of the likeliness that an event will occur. • Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty).

  19. Probability Theory • The probability P or Pr is 0 ≤ Pr ≤ 1 P = 0 (impossible event) P = 1 (sure event) • The higher the probability of an event, the more certain we are that the event will occur. A simple example is the toss of a fair coin. Since the two outcomes are equally probable, the probability of "heads" equals the probability of "tails", so the probability is 1/2 (or 50%) chance of either "heads" or "tails".

  20. Definition of Terms • Experiment– activity with observable result. • Trials – repetition of experiment. • Outcomes – results of each trial. • Sample Space – set of all possible outcomes. • Sample Points – elements of the sample space. • Event – a subset of the sample space. Example: Flipping a coin three times Write the sample space and an event A = getting exactly two heads.

  21. Probability of Simple Events • The probability of the occurrence of an event (called success) is Pr (E) = Where n = number of successes N = total number of possible outcomes • The probability of non-occurrence of an event (called failure) is Pr(E) = 1 – Pr (E)

  22. Examples • Tossing of a coin once S = {H, T} N = 2 outcomes E1 = getting a head on a single toss E1 = not getting a head (or getting a tail) Pr (E1) = Pr (E1) = 1 – =

  23. Examples • Tossing a pair of fair dice. Write the sample space and find the ff. • E1 = getting a double on a single toss • E1 = not getting a double on a single toss • A bag of marbles contains 4 red marbles and 6 black marbles. • What is the probability of getting a red marble in a single draw? • What is the probability of getting a black marble in a single draw? • What is the probability of getting a blue marble in a single draw?

  24. Marginal and Joint Probability • Marginal probability – is the probability of the occurrence of a single event or an event satisfying only one characteristic. • Joint probability – is the probability of two events occurring simultaneously in a single trial. Joint probability is also the probability of one event satisfying two or more characteristics. Let Pr(E1 ∩ E2) denote the joint probability of E1 and E2.

  25. Marginal and Joint Probability Example: If one card is picked from a deck of well shuffled cards, the probability of a king is a marginal probability; the probability of a card that is both a king and a heart is a joint probability.

  26. Mutually and Non-Mutually Exclusive Events • Two events E1 and E2 are mutually exclusive if it is impossible for both E1 and E2 to occur simultaneously in a single trial, that is , the joint probability of E1 and E2 is zero. • If E1 and E2 can occur simultaneously in a single trial, then they are not mutually exclusive. Example: In drawing one card from a deck of cards, the event of an ace and the event of a king are mutually exclusive while the event of an ace and the event of a heart are not mutually exclusive.

  27. The Addition Law • Given two events A and B; the probability that either A or B or both will occur is given by the sum of their respective probabilities. Pr(A U B) = Pr(A) + Pr(B) – Pr(A ∩ B) • Pr(A U B) stands for the probability that either A or B but not both A and B will occur. Pr(A ∩ B) stands for the joint probability of A and B.

  28. Example • One card is picked from a deck of well shuffled cards Let A = event of an ace K = event of a king H = event of a heart Find: • Probability of an ace or a king • Probability of an ace or a heart

  29. Conditional Probability • The probability that an event E2 will occur given that some event E1 has already occurred is called conditional probability, symbolized by Pr(E2 | E1). Pr(E2 | E1) =

  30. Conditional Probability Example • A biology student estimates that the probability that he will pass Biology II is 0.25, the probability that he will pass Chemistry I is 0.20, and the probability that he will pass both biology and chemistry is 0.10. What is the probability that the student will pass Chemistry I, given that he already passed Biology II?

  31. Independent and Dependent Events • Two events A and B are independent if the outcome of one event does not affect the likelihood of the other event. If A and B are independent, P(A | B) = P(A) and P(B | A) = P(B). • Two events A and B are dependent if the outcome of one event does affect the likelihood of the other event.

  32. The Multiplication Law • The probability that 2 events A and B will occur one after the other in the stated order is given by the product of their respective probabilities. For independent events (with replacement/repetition) Pr(A ∩ B) = Pr(A) • Pr(B) For dependent events (without replacement/repetition) Pr(A ∩ B) = Pr(A) • Pr(B | A)

  33. Example Three balls are drawn in succession from a box containing three red balls and four white balls. Let E1 = event that “1st ball is red” E2 = event that “2nd ball is white” E3 = event that “3rd ball is red” • If replacement is allowed, what is the probability that the balls drawn are in the order red, white, red? • If no replacement is allowed, what is the probability of drawing a red, a white and then a red ball alternately?

  34. Exercises • What is the probability of getting a 3 when a die is rolled? • What is the probability of getting a sum of two if two dice are tossed? • What is the probability of getting a total of 7 or 11 when a pair of dice is tossed? • Two marbles are drawn in succession from a box containing 10 red marbles and 10 blue marbles. What is the probability of drawing two blue marbles if replacement is not allowed?

  35. Exercises • A coin is flipped and a 6-sided die is rolled at the same time. What is the probability of flipping a head and rolling a 5? • A single 6-sided die is rolled. What is the probability of rolling a 2 or a 5? • A spinner has 4 equal sectors colored yellow, blue, green, and red. What is the probability of landing on red or blue after spinning this spinner? • A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a king or a club?

  36. Exercises • A glass jar contains 1 red, 3 green, 2 blue, and 4 yellow marbles. If a single marble is chosen at random from the jar, what is the probability that it is yellow or green? • In a math class of 30 students, 17 are boys and 13 are girls. On a unit test, 4 boys and 5 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student?

  37. Do Worksheet 8

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