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PHP 2510 Probability Part 1 Definition of probability measure • set notation and arithmetic • outcome, sample spaces, event Probability measures Important properties • Complements • Addition law PHP 2510 – Sept 15, 2008 1 Introduction Propellers of early probability theories included gamblers in the eighteenth century. Founders of modern probability theories include Pierre-Simon Laplace, Daniel Bernoulli, Carl F. Gauss, etc. The theory of probability has broad relevance in many areas. PHP 2510 – Sept 15, 2008 2 Examples: • Rhode Island reports 5 new cases of drug-resistant TB for the first quarter of some year. Does this represent an outbreak or epidemic? • An individual tests positive for HIV on a rapid screening test. What is the probability that the person actually is HIV-positive? • What is the likelihood of contracting HIV in one act of sexual intercourse? • What is the probability that someone with early stage breast cancer will survive for 10 years? Does the probability differ if she is 40 versus if she is 50? • If a graduate program makes 10 offers of admission, what is the expected number of students that will enroll? What is the probability that 10 will enroll? PHP 2510 – Sept 15, 2008 3 • What is the probability of winning Powerball with the purchase of one ticket? • For a woman who is pregnant with triplets, what is the probability of having three boys? • A statistics professor claims that he can distinguish between three varieties of cabernet. Given three unlabeled glasses, what is the probability he can classify them correctly just by guessing? • In a room with 25 people, what is the probability that two share the same birthday? • Consider a batter whose average is .333 (gets a hit once in every three at-bats). In a game where he has 4 at-bats, which of the following is more likely: zero hits or more than 2 hits? PHP 2510 – Sept 15, 2008 4 A little mathematics... A set is a collection of distinct objects (symbols, numbers, etc). • E.g. {red, blue, green}, {1, 3, 5, 11}. These distinct objects are called elements. The order in which the elements of a set are listed is irrelevant. • E.g. {red, blue, green} = {blue, green, red}. Special sets: R = {all real numbers}; Z = {all integers}; N ={all natural numbers}; ø is an empty set. PHP 2510 – Sept 15, 2008 5 Different ways to describe a set: • List all elements. E.g. {1, 3, 5, 2, 0}. • Use “...”. E.g. N = {1, 2, 3, ...}. • Use “:”. E.g. {m : m ∈ N and m > 8} is the same as {9, 10, 11, ...}, where the colon (“:”) means ”such that”. PHP 2510 – Sept 15, 2008 6 If every element of set A is also an element of set B, then A is said to be a subset of B, written A ⊆ B. • Graphically (Venn diagram), If A ⊆ B and B ⊆ A, then A = B. It is true vice versa. PHP 2510 – Sept 15, 2008 7 Basic operations: • The union of A and B, denoted by A ∪ B, is the set of all things which are elements of either A or B. Venn diagram, • The intersection of A and B, denoted by A ∩ B, is the set of all things which are elements of both A and B. Venn diagram, PHP 2510 – Sept 15, 2008 8 • A universal set U is a set that contains (as elements) all the entities one wishes to consider in a given situation. • The complement of A, denoted by Ā or Ac , is the set of all elements which are members of U , but not members of A Venn diagram, PHP 2510 – Sept 15, 2008 9 Sample spaces When thinking about the probability of observing a certain outcome that occurs at random, the first step is to figure out all possible outcomes. The set of all possible outcomes is called sample space, which we denote by Ω. Sample space is analogue to universal set. Example 1. Driving to work, you go through 3 intersections. At each intersection you either stop (S) or continue (C). Sample space is set of all possible outcomes Ω = {SSS, SSC, SCS, CSS, SCC, CSC, CCS, CCC} PHP 2510 – Sept 15, 2008 10 Example 2. A woman gives birth to twins. Sample space of possible gender combinations is Ω = {F F, M F, F M, M M } Example 3. A person attempts to quit smoking. The time from the beginning of his quit attempt until potential relapse is all times greater than zero. Ω = {t : t ∈ R and t > 0} Example 4. During the next year, you will have a certain number of visits to the doctor. The sample space is Ω = {0, 1, 2, . . .} PHP 2510 – Sept 15, 2008 11 Example 5. An individual takes a test for the HIV virus. The test result is either positive or negative, and his true HIV status is either HIV positive or HIV negative. Ω = {(T + H + ), (T + H − ), (T − H + ), (T − H − )} PHP 2510 – Sept 15, 2008 12 Event • Recall that sample space is all possible outcomes. • An event is a collection of outcomes. • In other words, an event is a subset of Ω. • The empty set ø can be an event. PHP 2510 – Sept 15, 2008 13 Union. A ∪ B is the event that either A or B occurs. Intersection. A ∩ B is the event that both A and B occur. Example 6 (twins). A = “one boy”, B = “no boy”. Then PHP 2510 – Sept 15, 2008 Ω = {F M, M F, M M, F F } A = {F M, M F } B = {F F } A∪B = {F M, M F, F F } A∩B = ø 14 Example 7 (twins). A = “first is a boy”. B = “at least one boy”. A = {M M, M F } B = {M M, M F, F M } A∪B = {M M, M F, F M } A∩B = {M M, M F } Example 8 (visits to the doctor). A = exactly 3 visits to the doctor, B = 2 or more visits to the doctor. PHP 2510 – Sept 15, 2008 Ω = {0, 1, 2, ...} A = {3} B = {2, 3, 4, . . .} A∪B = {2, 3, 4, . . .} A∩B = {3} 15 Complement. Ac or A is the event that A does not occur. Example 9 (twins). A = exactly one boy. A = {F M, M F } Ac = {M M, F F } Example 10 (visits to the doctor). B = 2 or more visits to the doctor. B = {2, 3, 4, . . .} Bc = {0, 1} Properties of complements: PHP 2510 – Sept 15, 2008 A ∪ Ac = Ω A ∩ Ac = ø 16 Probability measure Formally, it is a function labeled by P that maps subsets of the sample space Ω to the [0,1] interval. The probability measure P must satisfy the following axioms: 1. If an event A is in the sample space Ω, then P (A) ≥ 0 2. The probability of all possible events is one; i.e., P (Ω) = 1 3. If A and B are disjoint (cannot happen simultaneously; A ∩ B = ø), then P (A ∪ B) = P (A) + P (B). This also holds for more than two disjoint events. PHP 2510 – Sept 15, 2008 17 Two important properties Complements: P (Ac ) = 1 − P (A) Example 11: Suppose each combination of twins is equally likely (probability 0.25). A = ‘two boys’. Then A = {M M } P (A) = Ac P (Ac ) PHP 2510 – Sept 15, 2008 0.25 = {M F, F M, F F } = 1 − 0.25 = 0.75 18 Addition Law: P (A ∪ B) = P (A) + P (B) − P (A ∩ B) Example 12: Compute the probability that the first is a boy or the second is a boy. A = {M F, M M } B = {F M, M M } A∩B = {M M } P (A ∪ B) = P ({M F, M M }) + P ({F M, M M }) − P ({M M }) = 0.5 + 0.5 − 0.25 = 0.75 PHP 2510 – Sept 15, 2008 19 Summary Learning formal rules of probability is essential to understanding randomness and, eventually, statistics Probability measure maps subsets of a sample space to the unit interval. That is, it converts events in a sample space to probabilities that have values between 0 and 1. Become familiar with probability axioms. Properties of sets are important to computing probabilities (unions, intersections, complements). Become familiar with properties and laws governing sets – complement law, addition law, etc. PHP 2510 – Sept 15, 2008 20