932 CHAPTER 13 Counting and Probability 13.3 Assess Your Understanding 1. The Problem Discussed by Fermat and Pascal A game between two equally skilled players, A and B, is interrupted when A needs 2 points to win and B needs 3 points. In what proportion should the stakes be divided? (a) Fermat’s solution List all possible outcomes that can occur as a result of four more plays. Comparing the probabilities for A to win and for B to win then determines how the stakes should be divided. (b) Pascal’s solution Use combinations to determine the number of ways that the 2 points needed for A to win could occur in four plays. Then use combinations to determine the number of ways that the 3 points needed for B to win could occur. This is trickier than it looks, since A can win with 2 points in two plays, in three plays, or in four plays. Compute the probabilities, and compare them with the results in part (a). Historical Problem Although Girolamo Cardano (1501–1576) wrote a treatise on probability, it was not published until 1663 in Cardano’s collected works, and this was too late to have had any effect on the early development of the theory. In 1713, the posthumously published Ars Conjectandi of Jakob Bernoulli (1654–1705) gave the theory the form it would have until 1900. Recently, both combinatorics (counting) and probability have undergone rapid development, thanks to the use of computers. A final comment about notation. The notations ( ) C n r , and ( ) P n r , are variants of a form of notation developed in England after 1830. The notation ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n r for ( ) C n r , goes back to Leonhard Euler (1707–1783) but is now losing ground because it has no clearly related symbolism of the same type for permutations. The set symbols ∪ and ∩ were introduced by Giuseppe Peano (1858–1932) in 1888 in a slightly different context. The inclusion symbol ⊂ was introduced by E. Schroeder (1841–1902) about 1890. We owe the treatment of set theory in the text to George Boole (1815–1864), who wrote +A B for ∪A B and AB for ∩A B (statisticians still use AB for ∩A B ). 3. True or False The probability of an event can never equal 0. 4. True or False In a probability model, the sum of all probabilities is 1. 1. When the same probability is assigned to each outcome of a sample space, the experiment is said to have outcomes. 2. The of an event E is the set of all outcomes in the sample space S that are not outcomes in the event E. Concepts and Vocabulary 5. In a probability model, which of the following numbers could be the probability of an outcome? − 0 0.01 0.35 0.4 1 1.4 Skill Building 6. In a probability model, which of the following numbers could be the probability of an outcome? − 1.5 1 2 3 4 2 3 0 1 4 7. Determine whether the following is a probability model. Outcome Probability 1 0.2 2 0.3 3 0.1 4 0.4 8. Determine whether the following is a probability model. Outcome Probability Steve 0.4 Bob 0.3 Faye 0.1 Patricia 0.2 9. Determine whether the following is a probability model. Outcome Probability Shanice 0.3 Destiny 0.2 Jordan 0.1 Xavier 0.3 10. Determine whether the following is a probability model. Outcome Probability Kwamie 0.3 Joanne 0.2 Laura 0.1 Donna 0.5 Angela −0.1 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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