Question

7. Consider an experiment whose sample space consists of all positive integer (a.k.a. natural) numbers Z+ 1,2,3, ...J (i.e. choose a random natural number) a) Can you define a probability on Z+? (b) Can you define a probability on Z+ in such a way that any two numbers are equally likely to occur? (c) Along the lines of (b), can you define a probability on the interval [0, 1] in such a way that any two numbers in this interval are equally likely to occur?

Answer 1

Given Z= {1,2, 3 ...} = select a random natural number (a) Probability of any event E is the ratio of the favorable outcomes to the total number of outcomes The set Z+is a set of natural numbers and one number is selected at randomfrom this set As there is no bias present in Z+ each outcomes has an equal probability of occurrence Thus, the probability of selecting Favorable outcomes an outcome from Z+ is P(Z.) Total number of outcomes -p(say) Therefore, the probability on Z- isp (b) those events that have the same probability of occurrence. Equally likely events are As discussed above Z+ is a set of natural numbers anda random number is selected from this set As there is no bias present in the set Z4 (say like one of the observation is twice as likely to occur as the other) each outcomes has an equal probability of occurrence That is, every outcome is equally likely equally likely is p. Therefore, the probability on Z+ such that 2 numbers are (c) The interval is [0, 1], i.e. inclusive of 0 and 1

The natural number on this set is only 1 equally likely as there So it is not possible that two numbers, belonging to Z4, in this interval is only one natural number are Therefore, no we cannot define a probability on the interval [0, 1] in such a way that any two numbers in this interval are equally likely to occur