Stats - Assignment 3

Stats - Assignment 3 Case Study: Texas Hold’em (p. 222): Answer a, b, c, d, e, f, g. You must calculate results by hand (though you may use any technology of your choice to verify your answers). 222 CHAPTER 4 Probability Concepts student obtained. Simulate that experiment 1000 times. (Hint: expect a freshman to be selected? Compare that percentage 1 The simulation is equivalent to taking a random sample of with the actual percentage of the 1000 experiments in Which a size 1000 with replacement.) freshman was selected. (1. Referring to the simulation performed in part (c), in approx- e. Repeat parts (b)-(d) for sophomores; for juniors; for seniors. imately what percentage of the 1000 experiments would you fi CASE STUDY DISCUSSION TEXAS HOLD'EM At the beginning of this chapter on pages 156-157, we discussed Next recall that, after receiving your hole cards, there is a betting 1' Texas hold’em and described the basic rules of the game. Here round. Subsequently, 3 cards, called “the flop,” are dealt face up we examine some of the simplest probabilities associated with in the center of the table. To do the remaining problems, you need the game. to have studied either Sections 4.5 and 4.6 or Section 4.8. Assum- Recall that, to begin, each player is dealt 2 cards face down, ing that you are dealt a pocket pair, determine the probability that called “hole cards,” from an ordinary deck of 52 playing cards, as the flop '7- ili$;e:c;:,1:1eiefr:dilo 32%.: égiét::::f8t p OSSlble starting hand * (1. contains at least 1 card of your denomination. (Hint: Comple- mentation rule.) a. The probability that you are dealt pocket aces is 1/221, * e. gives you “trips,” that is, contains exactly 1 card of your or 0.00452 to three significant digits. If you studied either denomination and 2 other unpaired cards. Sections 4.5 and 4.6 or Section 4.8, verify that probability. * f. gives you “quads,” that is, contains 2 cards of your denomina- b. Using the result from part (a), obtain the probability that you tion. are dealt “pocket kings.” * g. gives you a “boat,” that is, contains 1 card of your denomina- c. Using the result from part (a) and your analysis in part (b), find tion and 2 cards of another denomination. the probability that you are dealt a “pocket pair,” that is, two cards of the same denomination. BIOGRAPHY i ANDREI KOLMOGOROV: FATHER OF MODERN PROBABILITY THEORY Andrei Nikolaevich Kolmogorov was born on April 25, 1903, in ancient to modern times and interpreted it in terms of dialec- Tambov, Russia. At the age of 17, Kolmogorov entered Moscow tical materialism, the philosophy originated by Karl Marx and State University, from which he graduated in 1925. His contributions Friedrich Engels. to the world of mathematics, many of which appear in his numerous Kolmogorov became a member of the faculty at Moscow articles and books, encompass a formidable range of subjects. State University in 1925 at the age of 22. In 1931, he was pro- Kolmogorov revolutionized probability theory with the in- moted to professor; in 1933, he was appointed a director of the troduction of the modern axiomatic approach to probability and Institute of Mathematics of the university; and in 1937, he became by proving many of the fundamental theorems that are a conse- Head of the University. quence of that approach. He also developed two systems of partial In addition to his work in higher mathematics, Kolmogorov differential equations that bear his name. Those systems extended was interested in the mathematical education of schoolchildren. the development of probability theory and allowed its broader He was chairman of the Commission for Mathematical Education application to the fields of physics, chemistry, biology, and under the Presidium of the Academy of Sciences of the USSR. civil engineering. During his tenure as chairman, he was instrumental in the devel- 1 In 1938, Kolmogorov published an extensive article entitled opment of a new mathematics training program that was intro- “Mathematics,” which appeared in the first edition of the Bo1- duced into Soviet schools. shaya Sovyetskaya Emsiklopediya (Great Soviet Encyclopedia). Kolmogorov remained on the faculty at Moscow State Uni- f In this article he discussed the development of mathematics from versity until his death in Moscow on October 20, 1987.