Chris Morgan, MATH G160 csmorgan@purdue.edu January 11, 2012 Lecture 2 Chapter 4.3: Probability Laws, DeMorgans Laws Chris Morgan, MATH G160 csmorgan@purdue.edu January 11, 2012 Lecture 2

Mutually Exclusive Two events A and B are said to be mutually exclusive, or disjoint, if their intersection is empty: A ∩ B = Two events A and B are said to be collectively exhaustive events if they are mutually exclusive but their union is the entire sample space, i.e. the events partition the space so that every outcome belongs to one and only one of the events. B A A B To say: such that but yet

DeMorgan’s Law (AUB)C= AC ∩ BC (A∩B)C= AC U BC

Let A, B, C be subsets of omega, Ω More Laws Let A, B, C be subsets of omega, Ω Distributive Laws: Associative and Commutative Laws:

Disjoint Sets and Pairwise Disjoint Two sets A and B are said to be disjoint sets if: This states that their intersection is the empty set, or that they have no elements in common. Also referred to as independent: Sets A1, A2, …, An are said to be pairwise disjoint if: whenever i ≠ j This implies that every pair is disjoint. Also referred to as mutually disjoint.

(Inclusion Exclusion Principle) Addition Law (Inclusion Exclusion Principle) P(AUB) = P(A) + P(B) –P(A∩B) – When the two events overlap, the intersection is counted twice; therefore, we subtract the interaction

IEP (cont) P(A U B U C) = P(A) + P(B) + P(C) – P(A∩B) – P(A∩C) – P(B∩C) + P(A∩B∩C) – Add all the “odd-way” intersections – Subtract all the “even-way” interactions

Example (I) - Cards Suppose we have a standard 52 card deck….

Example (II) Suppose we are in Las Vegas for an academic conference and are feely lucky. We go to Caesar’s Palace and try our hand at the casino game of craps. In this game we roll two fair six-sided dice and take the sum of the up faces to be the outcome of the game…

Example (III) Suppose little Johnny goes to the local video arcade where his three favorite video games are PacMac, Tetris, and Snake. Based on his past experience Johnny knows he can win PacMan 60% of the time, Tetris 50% of the time, and Snake 65% of the time. He further knows he will win both PacMac and Snake with probability 0.4, win both PacMan and Tetris with probability 0.2, and win both Tetris and Snake with probability 0.3, and will win all three games with probability 0.1. He decides to play each games once.

Example (IV) Suppose you are throwing a beach party and know that 100 guests will attend. - 65 will enjoy eating food. - 50 will enjoy hula dancing. - 60 will enjoy playing volleyball. - 10 will enjoy eating food and hula dancing but not playing volleyball. - 20 will enjoy eating food and playing volleyball but not hula dancing. - 20 will enjoy hula dancing and playing volleyball but not eating food. - 15 will enjoy eating food and hula dancing and playing volleyball. Draw the corresponding Venn diagram depicting the situation and label all of the pieces. How many people will enjoy at least one of the three activities? How many people will enjoy exactly two of the three activities?

Example (V) Suppose little Johnny goes to the local video arcade where his three favorite video games are PacMac, Tetris, and Snake. Based on his past experience Johnny knows he can win PacMan 60% of the time, Tetris 50% of the time, and Snake 65% of the time. He further knows he will win both PacMac and Snake with probability 0.4, win both PacMan and Tetris with probability 0.2, and win both Tetris and Snake with probability 0.3, and will win all three games with probability 0.1. He decides to play each games once.

Example (V) (cont) Probability he wins at least one game? Probability he wins exactly one game? Probability he wins no games? Probability he wins exactly two games? Probability he wins PacMan or Tetris? Probability he wins Tetris or Snake? Probability he wins at least two games?

Example (VI) The National Sporting Goods Association conducted a survey of persons 7 years of age or older about participation in sports activities (Statistical Abstracts in the U.S., 2002). The total population in this age group was reported at 248.5 million, with 120.9 million male and 127.6 million female. The number of participants (listed in millions) for the top five sports activities: Activity Male Prob Female Bicycling 22.2 21 Camping 25.6 24.3 Walking 28.7 57.7 Exercising 20.4 24.4 Swimming 26.4 34.4

Example (VI) (cont) a) For a randomly selected female, estimate the probability of participation in each of the sports activities. b) For a randomly selected male, estimate the probability of participation in each of the sports activities. c) For a randomly selected person, what is the probability the person participates in exercise walking? d) Suppose you just happen to see an exercise walker going by. What is the probability the walker is a woman? What is the probability the walker is a man?

Example (VII) A total of 28 percent of American males smoke cigarettes, 7 percent smoke cigars, and 5 percent smoke both cigarettes and cigars. a) What percentage of males smoke neither cigars nor cigarettes? b) What percentage smoke cigars but not cigarettes? c) What percentage smoke cigarettes but not cigars?

Example (VIII) High school seniors with strong academic records apply to the nation’s most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. The University of Pennsylvania received 2,851 applications for early admission. Of this group, it admitted 1,033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, Penn has admitted 18% of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2,375 (USA Today 01/24/2001). Let E, R, and D represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool, respectively.

Example (VIII) (cont) a) Use the data to estimate P(E), P(D), and P(R) . b) Are events E and D mutually exclusive? Find P(E D) . c) For the 2,375 students admitted to Penn, what is the probability that a randomly selected student was accepted during early admission? d) Suppose a student applies to Penn for early admission. What is the probability the student will be admitted for early admission or be deferred and later admitted during the regular admission process?

Example (IX) An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. These classes are open to any of the 100 students in the school. There are 28 students in the Spanish class, 26 in the French class, and 16 in the German class. There are 12 students that are in both Spanish and French, 4 that are in both Spanish and German, and 6 that are in both French and German. In addition, there are 2 students in all 3 classes. a) If a student is chosen randomly, what is the probability that he or she is not in any of these classes? b) If a student is chosen randomly, what is the probability that he or she is taking exactly one language class?

Example (X) What NCAA college basketball conferences have the higher probability of having a team play in college basketball’s national championship game? Over the last 20 years, the Atlantic Coast Conference (ACC) ranks first by having a team in the championship game 10 times. The Southeastern Conference (SEC) ranks second by having a team in the championship game 8 times. However, these two conferences have both had teams in the championship game only one time, when Arkansas (SEC) beat Duke (ACC) 76-70 in 1994 (NCAA website, April 2009). Use these data to estimate the following probabilities.

Example (X) (cont) a) What is the probability at least one team from these two conferences will be in the championship game? That is, what is the probability a team from the ACC or SEC will play in the championship game? b) What is the probability that the championship game will not have a team from one of these two conferences? c) What is the probability the championship game includes a team from the ACC but not the SEC? d) What is the probability the championship game includes a team from the SEC but not the ACC?

Example (XI) Two symmetric dice have both had two of their sides painted red, two painted black, one painted yellow, and the other painted white. When this pair of dice is rolled, what is the probability both land on the same color?

Example (XII) Box contains 3 marbles, 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box, the replacing it in the box and drawing a second marble from the box. Describe the sample space. Repeat when the second marble is drawn without first replacing the first marble.

Example (XIII) A 3-person basketball team consists of a guard, a forward, and a center. a) If a person is chosen at random from each of three different such teams, what is the probability of selecting a complete team? b) What is the probability that all 3 players selected play the same position?