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5 Problems 1. A white rook is placed at random on a chessboard. A black rook is then placed at random at one of the remaining squares. What is the probability that the black rook can capture the white rook? 58 CHAPTER 2. DISCRETE PROBABILITY 2. Jack and Jill are seated randomly at a pair of chairs at an 11 chair round table. What is the probability that they are seated next to each other? 3. In a 7 game playoff series between team A and team B, the first to win 4 games wins the series. 4. Team A plays the first 2 games at home, then the next 3 away and then 2 at home if necessary.

N, Ai ∩ Aj = φ for i = j} N P( N Ai ) = i=1 P (Ai ) i=1 Proof. Consider the countable sequence of events {Ai : i = 1, 2, . . , ∞, Ai ∩ Aj = φ for i = j} where Ai = φ for i = N + 1, N + 2, . . , ∞ we have N Ai ) = P ( P( i=1 ∞ Ai ) = i=1 ∞ N P (Ai ) P (Ai ) = i=1 i=1 and finite additivity holds for disjoint events. 3 P (Ac ) = 1 − P (A). Proof. Using S = A ∪ Ac , where A ∩ Ac = φ, and finite additivity 1 = P (S) = P (A ∪ Ac ) = P (A) + P (Ac ) . Subtracting P (A) from both sides gives the result.

Calculate the 3-trimmed mean. 6. Calculate the 5-Winsorized mean. 7. Write a computer program to calculate the Walsh sum median. 8. Calculate the sample range R. 9. Calculate the inter quartile range IQR. 10. Calculate the mean absolute deviation about the median D. M. and the R Development Core Team (2004). An Introduction to R. org 36 CHAPTER 1. DESCRIPTIVE STATISTICS 11. Calculate the sample variance S 2 . 12. 4). 13. 1. 4 4 6 10 12 5 14. Calculate the sample median. 15. Calculate the Walsh sum median.

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A computational approach to statistics by Klotz J.H.

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