Chapter 5 - Elementary Probability Theory

Transcription

1 Chapter 5 - Elementary Probability Theory Historical Background Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling was Pierre Simon de Laplace, who is often credited with being the father of probability theory. In the twentieth century a coherent mathematical theory of probability was developed through people such as Chebyshev, Markov, and Kolmogorov. Probability The study of probability is concerned with random phenomena. Even though we cannot be certain whether a given result will occur, we often can obtain a good measure of its likelihood, or probability. In the study of probability, any observation, or measurement, of a random phenomenon is an experiment. The possible results of the experiment are called outcomes, and the set of all possible outcomes is called the sample space. Usually we are interested in some particular collection of the possible outcomes. Any such subset of the sample space is called an event. Example: Tossing a Coin If a single fair coin is tossed, find the probability that it will land heads up. Definition: A phenomenon is Random if individual outcomes are uncertain cannot be determined beforehand but there is a regular distribution of outcomes in a large number of repetitions of the experiment. Probability is the relative frequency of an outcome over a long series of repetitions of the event. Probability is a numerical measure between 0 and 1 that describes the likelihood that an event will occur 1 P a g e

2 Probability Probability is a numerical measure that indicates the likelihood of an event. All probabilities are between 0 and 1, inclusive. A probability of 0 means the event is impossible. A probability of 1 means the event is certain to occur. Events with probabilities near 1 are likely to occur. Events can be named with capital letters: A, B, C P(A) means the probability of A occurring. P(A) is read P of A 0 P(A) 1 P(A) = 1, the event is certain to occur P(A) = 0, the event is certain not to occur Theoretical Probability Formula If all outcomes in a sample space S are equally likely, and E is an event within that sample space, then the theoretical probability of the event E is given by number of favorable outcomes ne ( ) PE ( ). total number of outcomes ns ( ) Example - Among a sample of 50 dog owners, 23 feed their dogs Mighty Mutt dry dog food. Estimate the probability that a dog owner selected at random feeds their dogs Mighty Mutt dry food. a). 23/50 b). 27/50 c). 1/23 d). 23/27 Example: Flipping a Cup A cup is flipped 100 times. It lands on its side 84 times, on its bottom 6 times, and on its top 10 times. Find the probability that it will land on its top. 2 P a g e

3 Empirical Probability Formula If E is an event that may happen when an experiment is performed, then the empirical probability of event E is given by number of times event E occurred PE ( ). number of times the experiment was performed Example: Card Hands There are 2,598,960 possible hands in poker. If there are 36 possible ways to have a straight flush, find the probability of being dealt a straight flush. Example: Gender of a Student A school has 820 male students and 835 female students. If a student from the school is selected at random, what is the probability that the student would be a female? Law of Large Numbers As an experiment is repeated more and more times, the proportion of outcomes favorable to any particular event will tend to come closer and closer to the theoretical probability of that event. Example: Toss a coin repeatedly. The relative frequency gets closer and closer to P(head) = 0.50 Relative Frequency n = number of heads f = number of flips P a g e

4 Probability in Genetics Gregor Mendel, an Austrian monk used the idea of randomness to establish the study of genetics. To study the flower color of certain pea plants he found that: Pure red crossed with pure white produces red. Mendel theorized that red is dominant (symbolized by R), while white is recessive (symbolized by r). The pure red parent carried only genes for red (R), and the pure white parent carried only genes for white (r). Every offspring receives one gene from each parent which leads to the tables below. Every second generation is red because R is dominant. Example: Probability of Flower Color Referring to the 2 nd to 3 rd generation table (previous slide), determine the probability that a third generation will be a) red b) white Base the probability on the sample space of equally likely outcomes: S = {RR, Rr, rr, rr}. 4 P a g e

5 Intersection of Sets The intersection of sets A and B, written A B, is the set of elements common in both A and B, or Union of Sets The union of sets A and B, written A B, is the set of elements belonging to wither of the sets, or Disjoint Sets Two sets have no elements in common, sets A and B are disjoint if A B = ø. Meaning of A and B - Intersection of Sets AND The Venn diagram to the left shows two sets A and B the intersection of the two sets is the gray portion,, U is the universe Meaning of A or B Union of Sets OR The Venn diagram to the left shows two sets A and B the union of the two sets is the gray portion,, U is the universe 5 P a g e

6 Events Involving Not and Or Properties of Probability Let E be an event from the sample space S. That is, E is a subset of S. Then the following properties hold PE ( ) 1 (The probability of an event is between 0 and 1, inclusive.) 2. P( ) 0 (The probability of an impossible event is 0.) 3. PS ( ) 1 (The probability of a certain event is 1.) Example: Rolling a Die When a single fair die is rolled, find the probability of each event. a) the number 3 is rolled b) a number other than 3 is rolled c) the number 7 is rolled d) a number less than 7 is rolled Events Involving Not The table on the next slide shows the correspondences that are the basis for the probability rules developed in this section. For example, the probability of an event not happening involves the complement and subtraction. 6 P a g e

7 Correspondences Set Theory Logic Arithmetic Operation or Connective (Symbol) Operation or Connective (Symbol) Operation or Connective (Symbol) Complement Not Subtraction Union Or Addition Intersection And Multiplication Probability of a Complement The probability that an event E will not occur is equal to one minus the probability that it will occur. P(not E) P( S) P( E) 1 PE ( ) So we have and P( E) P E 1 P( E) 1 P( E ). Example: Complement When a single card is drawn from a standard 52-card deck, what is the probability that is will not be an ace? 7 P a g e

8 Events Involving Or Probability of one event or another should involve the union and addition. Mutually Exclusive Events Two events A and B are mutually exclusive events if they have no outcomes in common. (Mutually exclusive events cannot occur simultaneously.) Addition Rule of Probability (for A or B) If A and B are any two events, then P( A or B) P( A) P( B) P( A and B). If A and B are mutually exclusive, then P( A or B) P( A) P( B). Example: Probability Involving Or When a single card is drawn from a standard 52-card deck, what is the probability that it will be a king or a diamond? Example: Probability Involving Or If a single die is rolled, what is the probability of a 2 or odd? 8 P a g e

9 Probability versus Statistics Probability is the field of study that makes statements about what will occur when a sample is drawn from a known population. Statistics is the field of study that describes how samples are to be obtained and how inferences are to be made about unknown populations. Conditional Probability; Events Involving And Conditional Probability Sometimes the probability of an event must be computed using the knowledge that some other event has happened (or is happening, or will happen the timing is not important). This type of probability is called conditional probability. The probability of event B, computed on the assumption that event A has happened, is called the conditional probability of B, given A, and is denoted P(B A). Example: Selecting From a Set of Numbers From the sample space S = {2, 3, 4, 5, 6, 7, 8, 9}, a single number is to be selected randomly. Given the events A: selected number is odd, and B selected number is a multiple of 3. find each probability. a) P(B) b) P(A and B) c) P(B A) 9 P a g e

10 Conditional Probability Formula The conditional probability of B, given A, and is given by P( A B) P( A and B) P( B A). P( A) P( A) Example: Probability in a Family Given a family with two children, find the probability that both are boys, given that at least one is a boy. Mutually Exclusive Events Two events are mutually exclusive if they cannot occur at the same time. Mutually Exclusive = Disjoint If A and B are mutually exclusive, then P(A and B) = 0 Addition Rules If A and B are mutually exclusive, then P(A or B) = P(A) + P(B). If A and B are not mutually exclusive, then P(A or B) = P(A) + P(B) P(A and B). 10 P a g e

11 Independent Events Two events A and B are called independent events if knowledge about the occurrence of one of them has no effect on the probability of the other one, that is, if P(B A) = P(B), or equivalently P(A B) = P(A). P(A B) denotes the probability that event A will occur given that event B has occurred. This is called conditional probability. Read Probability of A given B. Two event s A and B are called dependent events if the occurrence of event B has changed the probability that event A will occur, that is Example: Checking for Independence A single card is to be drawn from a standard 52-card deck. Given the events A: the selected card is an ace B: the selected card is red a) Find P(B). b) Find P(B A). c) Determine whether events A and B are independent. Multiplication Rules 11 P a g e

12 Example: Multiplication, Independent Events Suppose you are going to throw two fair die. What is the probability of getting a 5 on each die? Example: Selecting From an Jar of Balls, Dependent Events Jeff draws balls from the jar below. He draws two balls without replacement. Find the probability that he draws a red ball and then a blue ball, in that order. 4 red 3 blue 2 yellow 12 P a g e

13 Example: Selecting From an Jar of Balls, Independent Example Jeff draws balls from the jar below. He draws two balls, this time with replacement. Find the probability that he gets a red and then a blue ball, in that order. 4 red 3 blue 2 yellow Critical Thinking Pay attention to translating events described by common English phrases into events described using and, or, complement, or given. Rules and definitions of probabilities have extensive applications in everyday lives. 13 P a g e

14 Multiplication Rule for Counting This rule extends to outcomes involving three, four, or more series of events. Example - A coin is tossed and a six-sided die is rolled. How many outcomes are possible? a). 8 b). 10 c). 12 d). 18 Tree Diagrams Displays the outcomes of an experiment consisting of a sequence of activities. The total number of branches equals the total number of outcomes. Each unique outcome is represented by following a branch from start to finish. Tree Diagrams with Probability We can also label each branch of the tree with its respective probability. To obtain the probability of the events, we can multiply the probabilities as we work down a particular branch. 14 P a g e

15 Example - Place five balls in an urn: three red and two blue. Select a ball, note the color, and, without replacing the first ball, select a second ball. The Factorial For any counting number n, the quantity n factorial is given by Example Evaluating Expressions Containing Factorials Evaluate each expression. a) 3! b) 6! c) (6-3)! d) 6!-3! Arrangements of n Distinct objects n! Te total number of different ways to arrange n distinct objects is n! Example Arranging Essays Erika Berg has seven essays to include in her English 1A folder. In how many different orders can she arrange them? 15 P a g e

16 Permutations - arrangements are often called permutations, the number of permutations of n distinct things taken r at a time is denoted np r Since the number of objects being arranged cannot exceed the total number available, we assume for our purposes here that r n. Applying the fundamental counting principle to arrangements of this type gives: np r = Permutations are to evaluate the number of arrangements of n things taken r at a time, where repetitions are not allowed, and the order of the items is important. Example - For a group of seven people, how many ways can four of them be seated in four chairs? a). 35 b). 3 c). 28 d). 840 Alternative Notations are P(n,r) and For Example 4P 2 means the number of permutations of 4 distinct things taken 2 at a time. Using a graphing calculator we can perform this calculation directly as follows for a TI-83: For 10P 6 enter in 10 then hit MATH Scroll over to PRB and scroll down to 2 (npr) hit enter then enter in 6 then hit enter. 10P 6 = P a g e

17 Example Using the Factorial Formula for Permutations Evaluate each permutation a) 4P 2 b) 8P 5 Note that 5P 5 is equal to 5! It is true for all whole numbers n that np n = n! Combinations - the number of combinations of n things taken r at a time (that is the number of size r subsets, given a set of size n) is written nc r Since there are n things available and we are choosing r of them, we can read nc r as n choose r. The formula for evaluating numbers of combinations nc r Permutations are to evaluate the number of arrangements of n things taken r at a time, where repetitions are not allowed, and the order of the items is important. Combinations are the number of combinations of n things taken r at a time (that is the number of size r subsets, given a set of size n), where repetitions are not allowed, and the order is not important. Factorial Formula for Combinations The number of combinations, or subsets, of n, distinct things taken r at a time, where r n, can be calculated as nc r Alternative Notations are C(n,r) and and 17 P a g e

18 Using a graphing calculator we can perform this calculation directly as follows for a TI-83: For 14C 6 enter in 14 then hit MATH Scroll over to PRB and scroll down to 2 (ncr) hit enter then enter in 6 then hit enter. 14C 6 = 3003 Example Evaluate each combination a) 9C 7 b) 24C 18 Example - Among eleven people, how many ways can eight of them be chosen to be seated? a). 6,652,800 b). 165 c). 3 d) P a g e