Here are the problems written out, in case you don't have the text:

• 1.19: A coin is to be tossed as many times as necesary to turn up one head. Thus the elements, c, of the sample space S are H, TH, TTH, TTTH, and so forth. Let the probability set function P assign to these elements the respective probabilities 1/2, 1/4, 1/8, 1/16, and so forth. Show that P(S) = 1. Let C_1 = {c: c is H, TH, TTH, TTTH, or TTTTH}. Compute P(C_1). Let C_2 = {c: c is TTTTH or TTTTTH}. Compute P(C_2), P(C_1 and C_2), and P(C_1 or C_2).
• 1.23: If C_1 and C_2 are subsets of the sample space S, show that P(C_1 and C_2) <= P(C_1) <= P(C_1 or C_2) <= P(C_1) + P(C_2)
• 1.36: A drawer contains eight pairs of socks. If six socks are taken at random and without replacement, compute the probability that there is at least one matching par among these six socks. Hint: Compute the probability that there is not a matching pair.
• 1.38: Bowl I contains 3 red chips and 7 blue chips. Bowl II contains 6 red chips and 4 blue chips. A bowl is selected at random and then 1 chip is drawn from this bowl.
  (a) Compute the probability that this chip is red.
  (b) Relative to the hypothesis that the chip is red, find the conditional probaility that it is drawn from bowl II.
• 1.47: Let a card be selected from an ordinary deck of playing cards. The outcome, c, is one of these 52 cards. Let X(c)=4 if c is an ace, X(c)=3 if c is a king, X(c)=2 if c is a queen, X(c)=1 if c is a jack, and X(c)=0 otherwise. Suppose P assigns a probability of 1/52 to each outcome c. Describe the induced probability P_x(A) on the space S = {0, 1, 2, 3, 4} of the random variable X.
• 1.65: Let f(x) = 1/(x^2), 1 < x < infinity, zero elsewhere, be the p.d.f. of X. If A_1 = {x: 1 < x < 2} and A_2 = {x: 4 < x < 5}, find P(A_1 or A_2) and P(A_1 and A_2).
• 1.66: A mode of a distribution of one random variable X is a value of x that maximizes the p.d.f. f(x). For X of the continuous type, f(x) must be continuous. If there is only one such x, it is called the mode of the distribution. Find the mode of each of the following distributions:
  (a) f(x) = (1/2)^x, x=1,2,3,..., zero elsewhere
  (b) f(x) = 12x^2(1-x), 0 < x < 1, zero elsewhere
  (c) f(x) = (1/2)x^2 exp(-x), 0 < x < infinity, zero elsewhere
• 1.76: Let X have p.d.f. f(x) = 4x^3, 0 < x < 1, zero elsewhere. Find the distribution function and p.d.f. of Y = -2 ln(X^4).
• 1.84: Let X have p.d.f. f(x) = 3x^2, 0 < x < 1, zero elsewhere. Consider a random rectangle whose sides are X and (1-X). Determine the expected value of the area of the rectangle.
• 1.90: Let f(x) = (1/2)^x, x = 1,2,3,..., zero elsewhere, be the p.d.f. of the random variable X. Find the m.g.f., the mean, and the variance of X.