MA : Introductory Probability

Transcription

1 MA : Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky Spring 2017 David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

2 Probability Games. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

3 Probability Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes relative likelihoods and distributions. In common usage, the word "probability" is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%. The analysis of events governed by probability is called statistics. From Wolfram MathWolrd. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

4 What are we going to learn this week? Sample Space. Event. Random variable. Distribution function. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

5 Notations Z: the set of integers. N: the set of natural numbers. Z + : the set of positive integers. Q: the set of rational numbers. Q : the set of nonzero rational numbers. R: the set of real numbers. C: the set of complex numbers. For each n Z +, Z n = {0, 1, 2,..., n}. [a, b] = {x R : a x b}. (a, b) = {x R : a < x < b}. (a, b] = {x R : a < x b}. [a, b) = {x R : a x < b}. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

6 Set Operations and the Laws of Set Theory Definition For A, B U, The union of A and B, The intersection of A and B, A B = {x : x A or x B} A B = {x : x A and x B} The symmetric difference of A and B, A B = {x : x A B and x / A B} David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

7 Set Operations and the Laws of Set Theory Definition Let A, B U. The sets A and B are called disjoint when A B =. Theorem If A, B U. Then A and B are disjoint if and only if A B = A B. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

8 Set Operations and the Laws of Set Theory Definition For A U, the complement of A, denoted U A, or A, is given by {x : x U and x / A} Definition For A, B U, the (relative) complement of A in B, denoted B A, is given by {x : x B and x / A} David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

9 Set Operations and the Laws of Set Theory Theorem For A, B U, the following statements are equivalent: 1 A B 2 A B = B 3 A B = A 4 B A David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

10 The Laws of Set Theory For A, B, C U, 1 Law of double complement: A = A. 2 DeMorgan s Laws: A B = A B A B = A B. 3 Commutative Laws: A B = B A and A B = B A. 4 Associative Laws: A (B C) = (A B) C A (B C) = (A B) C. 5 Distributive Laws: A (B C) = (A B) (A C) A (B C) = (A B) (A C). 6 Idempotent Laws: A A = A and A A = A. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

11 Counting and Venn Diagrams Rule If A and B are finite sets, then A B = A + B A B Consequently, A and B are disjoint if and only if A B = A + B. In addition, when U is finite, from the DeMorgan s law we have A B = A B = U A B = U A B + A B. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

12 Counting and Venn Diagrams Rule If A, B, C are finite sets, then A B C = A + B + C A B A C B C + A B C In addition, when U is finite, from the DeMorgan s law we have A B C = A B C = U A B C = U A B C + A B + A C + B C A B C. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

13 A First Word on Probability: Sample Space Definition Given an experiment such as tossing a single fair coin, rolling a single fair die, or selecting two students at random from a class of 20, a set of all possible outcomes for each situation is called a sample space or outcome space. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

14 A First Word on Probability: Event Definition Under the assumption of equal likelihood, let ϕ be the sample space for an experiment ξ. Each subset A ϕ, including the empty subset, is called an event. Each element of ϕ determines an outcome, so if ϕ = n and a ϕ, A ϕ, then P(a) = the probability that a occurs = a ϕ = 1 n. P(A) = the probability that a occurs = A ϕ = A n. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

15 Examples Example When Daphne tosses a fair coin, what is the probability she gets a head? David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

16 Examples Example When Daphne tosses a fair coin, what is the probability she gets a head? ϕ = {H, T } and a = H so P(a) = a ϕ = 1 2 David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

17 Examples Example If John rolls a fair die, what is the probability he gets 1 a 5 or a 6? 2 an even number? David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

18 Examples Example If John rolls a fair die, what is the probability he gets 1 a 5 or a 6? 2 an even number? For (1) ϕ = {1, 2, 3, 4, 5, 6} and A = {5, 6} so P(A) = A ϕ = 2 6 = 1 3 David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

19 Examples Example If John rolls a fair die, what is the probability he gets 1 a 5 or a 6? 2 an even number? For (1) ϕ = {1, 2, 3, 4, 5, 6} and A = {5, 6} so For (2) so P(A) = A ϕ = 2 6 = 1 3 ϕ = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6} P(A) = A ϕ = 3 6 = 1 2 David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

20 The cartesian product Definition For sets A, B, the cartesian product of A and B is A B = {(a, b) : a A, b B} The elements of A B are called ordered pairs. For (a, b), (c, d) A B, we have (a, b) = (c, d) if and only if a = c and b = d. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

21 Examples Example If A = {1, 2, 3} and B = {x, y}, then A B = {(1, x), (1, y), (2, x), (2, y), (3, x), (3, y)} B A = {(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3)} So Although A B B A A B = A B = 6 = B A = B A David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

22 Examples Example Suppose Rosa rolls two fair dice. This experiment can be decomposed as follows. Let ξ 1 be the experiment where the first die is roll - with sample space ϕ 1 = {1, 2, 3, 4, 5, 6}. Likewise we let ξ 2 be the experiment where the second die is rolled - also with sample space ϕ 2 = {1, 2, 3, 4, 5, 6}. Consequently, when Rosa rolls these dice the sample space is ϕ = ϕ 1 ϕ 2 = {(x, y) : x, y = 1, 2, 3, 4, 5, 6} What is the probability she gets rolls a 6? That is, the top faces of the dice sum to 6. David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22

23 Examples Example Suppose Rosa rolls two fair dice. This experiment can be decomposed as follows. Let ξ 1 be the experiment where the first die is roll - with sample space ϕ 1 = {1, 2, 3, 4, 5, 6}. Likewise we let ξ 2 be the experiment where the second die is rolled - also with sample space ϕ 2 = {1, 2, 3, 4, 5, 6}. Consequently, when Rosa rolls these dice the sample space is ϕ = ϕ 1 ϕ 2 = {(x, y) : x, y = 1, 2, 3, 4, 5, 6} What is the probability she gets rolls a 6? That is, the top faces of the dice sum to 6. Solution: A = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)} with Pr(A) = A ϕ = David Murrugarra (University of Kentucky) MA 320: Section 1.1 Spring / 22