Robert Bartoszynski - Probability and Statistical Inference

Either Paul tosses more heads than Peter (event ) or he tosses more tails than Peter (event ). These two events exclude one another and exhaust all possibilities (since one cannot have ties in number of heads and number of tails). Switching the role of heads and tails transforms one of these events into the other. Thus, sample space becomes partitioned into two equiprobable events, and we must have .

The use of ( 2.7) requires techniques for counting the numbers of elements in some sets. These topics, known under the name combinatorics , will be discussed in Chapter 3.

1 2.5.1 A coin is tossed seven times. Assume that each of the possible outcomes (sequences like HTTHHTH of length 7) is equally likely. Relate each outcome to a binary number by replacing H by 1 and T by 0, for example, THHHTTH is 0111001 = 57. Find the probability that a number generated in this way lies between 64 and 95 (inclusive on both sides).

2 2.5.2 A number is chosen at random from the series 4, 9, 14, 19, …, and another number is chosen from the series 1, 5, 9, 13, …. Each series has 100 terms. Find .

3 2.5.3 A regular die and a die with 2, 3, 5, 6, 7, and 8 dots are tossed together, and the total number of dots is noted. What is the probability that the sum is greater than or equal to 10?

4 2.5.4 Use formula ( 2.6) to find the number of primes not exceeding 100. [Hint: Assume that you sample one of the numbers 1, 2, …, 100. Let be the event “the number sampled is divisible by .” Determine . Then the answer to the problem is (why?).]

Looking at Axiom 3, one may wonder why do we need it for the case of countable (and not just finite) sequences of events. Indeed, the necessity of all three axioms, with only finite additivity in Axiom 3, can be easily justified simply by using probability to represent the limiting relative frequency of occurrences of events. Recall the symbol from Section 2.1for the number of occurrences of the event in the first experiments. The nonnegativity axiom is simply a reflection of the fact that the count cannot be negative. The norming axiom reflects the fact that event is certain and must occur in every experiment so that , and hence, . Finally, (taking the case of two disjoint events and ), we have , since whenever occurs, does not, and conversely. Thus, if probability is to reflect the limiting relative frequency, then should be equal to , since the frequencies satisfy the analogous condition .

The need for countable additivity, however, cannot be explained so simply. This need is related to the fact that to build a sufficiently powerful theory, one needs to take limits. If is a monotone sequence of events (increasing or decreasing, i.e., or ) then , where the event has been defined in Section 1.4. Upon a little reflection, one can see that such continuity is a very natural requirement. In fact, the same requirement has been taken for granted for over 2,000 years in a somewhat different context: in computing the area of a circle, one uses a sequence of polygons with an increasing number of sides, all inscribed in the circle. This leads to an increasing sequence of sets “converging” to the circle, and therefore the area of the circle is taken to be the limit of the areas of approximating polygons. The validity of this idea (i.e., the assumption of the continuity of the function = area of ) was not really questioned until the beginning of the twentieth century. Research on the subject culminated with the results of Lebesgue.

To quote the relevant theorem, let us say that a function , defined on a class of sets (events), is continuous from below at the set if the conditions and imply that . Similarly, is continuous from above at the set if the conditions and imply . A function that is continuous at every set from above or from below is simply called continuous (above or below). Continuity from below and from above is simply referred to as continuity .