5) Chapter 14 # 29: Pascal stated that the odds in favor of throwing a six in four throws of a single die are 671 to 625. Show why this is true, using Huygen's method: E(X) = (p/r)a + (q/r)b found on page 496 (not the geometric distribution formula). (3 points) 6) Chapter 14 # 32: Suppose three players play a fair series of games under the condition that the first player to win three games wins the stakes. If they stop play when the first player needs one game while the second and third players each need two games, find the fair division of the stakes. (This problem was discussed in the correspondence of Pascal with Fermat.) (Hint: draw a tree diagram for all possible sequences of outcomes. Since there could be up to 3 games remaining, there are 27 possible sequences, though some would be truncated if a player wins the stakes before all possible games need to be played.) (3 points) 7) Chapter 14 # 36: There are 12 balls in an urn, 4 of which are white and 8 black. Three blindfolded players, A, B, C draw a ball in turn, first A, then B, then C. The winner is the one who first draws a white ball. Assuming that each (black) ball is replaced after being drawn, find probability of winning for each player. Keep in mind that player A could win on the first draw, or the 4*, or the 7*", or the 10" and so forth. The probabilities of winning on these respectively draws forms an infinite geometric sequence. (3 points) 8) Chapter 15, Question 4 (modified): Use Fermat's method of setting bx; x13 = bx, xg, solving for b and then setting x = x; = x; to_find the maximum of bxx3. (3 points) 9) Chapter 15, Question 6: Use Fermat's tangent method to determine the relation between the abscissa x of a point B and the subtangent t that gives the tangent line to y = x3. (3 points) . t a) First, set % = ? and solve for subtangent t as e approaches zero. b) Once you have t, find the slope of the tangent line by computing f(x)/t. 10) Chapter 15, question 12: Use Hudde's rule applied to Descartes' method to show that the slope of the tangent line to y = x at (x, x) is nx"1. Descartes' method gives us yZ + (x v)% = n?, a) Replace y with x and expand (x v)? to x2 2vx + v2. Apply Hudde's rule (with p=0and b = 1) to find a simpler equation: applying Hudde's rule to f(x) = ag + a;Xx + a,x? + - + @, x" gives g(xX) = a,;x + 2a,x% + -+ na,x". (4 points) b) Rearrange g(x) to solve for v x. n = SO the slope of the tangent line can be found by computing ];;: c) The slope of the normalism =