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If the sample space is  and if  is a set for which the integral  exists, show that this set function is not a probability set function. What constant do we multiply the integrand by to make it a probability set function?

Compute the probability of being dealt at random and without replacement a 13-card bridge hand consisting of: (a) 6 spades, 4 hearts, 2 diamonds, and 1 club; (b) 13 cards of the same suit. Three distinct integers are chosen at random from the first 20 positive integers. Compute the probability that: (a) their sum is

A person has purchased 10 of 1000 tickets sold in a certain raffle. To determine the five prize winners, five tickets are to be drawn at random and without replacement. Compute the probability that this person wins at least one prize. Hint: First compute the probability that the person does not win a prize.

A bowl contains 16 chips, of which 6 are red, 7 are white, and 3 are blue. If four chips are taken at random and without replacement, find the probability that: (a) each of the four chips is red; (b) none of the four chips is red; (c) there is at least one chip of

Answer these questions only from chapter 14! One page for each question, total of four pages. 1) Explain and discuss the five stages of identity development. In what ways might understanding these stages be helpful in working with a client? 2) Explain and discuss the five stages of white identity development. In what ways might

1. A secretary types three letters and the three corresponding envelopes. In a hurry, he places at random one letter in each envelope. What is the probability that at least one letter is in the correct envelope? 2. Consider poker hands drawn from a well-shuffled deck as described in Example 1.3.4. Determine the probability of

Consider the events  For the last two exercises it is assumed that the reader is familiar with σ-fields.

Suppose the experiment is to choose a real number at random in the interval (0, 1). For any subinterval (a, b) ⊂ (0, 1), it seems reasonable to assign the probability P[(a, b)] = b−a; i.e., the probability of selecting the point from a subinterval is directly proportional to the length of the subinterval. If

Prove expression (1.3.9). Consider Remark 1.3.2. (b) Now prove the general inclusion exclusion formula given by the expression (1.3.13). 1.3.13 Three distinct integers are chosen at random from the first 20 positive integers. Compute the probability that: (a) their sum is even; (b) their product is even.

Suppose D is a nonempty collection of subsets of C. Consider the collection of events Note that φ ∈ B because it is in each σ-field, and, hence, in particular, it is in each σ-field . Continue in this way to show that B is a σ-field.