Название: Probability and Statistical Inference
Автор: Robert Bartoszynski
Издательство: John Wiley & Sons Limited
Жанр: Математика
isbn: 9781119243823
isbn:
In many real situations, the outcomes in the sample space reveal a certain symmetry, derived from physical laws, from logical considerations, or simply from the sampling scheme used. In such cases, one can often assume that the outcomes are equiprobable and use (2.7) as a rule for computing probabilities. Obviously, the function
To use some very simple examples, in tossing a regular die, each face has the same probability
The case above is rather trivial, but considerations of symmetry can sometimes lead to unexpectedly simple solutions of various problems.
Example 2.6
Peter tosses a fair coin
Solution
Either Paul tosses more heads than Peter (event
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.
Problems
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?).]
2.6 Necessity of the Axioms*
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
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