Which of the following describes an example of independent events?

A.
generating a random number between 1 and 25, and then generating another random number between 1 and 25, excluding the first one picked
B.
generating a random number between 1 and 25, and then generating another random number between 1 and 25, including the first one picked
C.
generating a random number between 2 and 25, and then generating another random number less than the first one picked
D.
generating a random number between 1 and 24, and then generating another random number greater than the first one picked

Question

Which of the following describes an example of independent events?

A.
generating a random number between 1 and 25, and then generating another random number between 1 and 25, excluding the first one picked
B.
generating a random number between 1 and 25, and then generating another random number between 1 and 25, including the first one picked
C.
generating a random number between 2 and 25, and then generating another random number less than the first one picked
D.
generating a random number between 1 and 24, and then generating another random number greater than the first one picked

aakashsudhakar · Answer

Think of it this way. Independent events suggest that the occurrence of the second event is absolutely independent of the occurrence of the first event. In other words, absolutely NOTHING that the first event accomplishes has ANY bearing on the occurrence of the second events. 

One way to view this is that if the total possible outcomes that can be obtained in the first event is given by X and the total possible outcomes that can be obtained in the second event is given by Y, then the following relationships are true: $$X = \left\{ S _{1} \right\}; Y = \left\{ S _{2} \right\}$$where S1 is the sample size of the total number of possible outcomes in the first event and S2 is the sample size of the total number of possible outcomes in the second event. 

Now let's consider the experiment in question. The experimenter is performing something simple: randomly generating one number value from 1 to 25. If you notice, in every experiment, regardless of the particular conditional constraints that vary per choice, both the first and second experiment suggest a theoretical total sample size from 1 to 25, since both suggest drawing a value from a theoretically total pool of integers from 1 to 25.

Logically, it follows that if the first and second experiment are independent, then the second experiment and the first experiment should have equal sample sizes, from 1 to 25, suggesting that the occurrence and outcome of the first experiment had absolutely no bearing on the occurrence and outcome of the second experiment. 

Only one choice amidst the four possibles fulfills that criterion. Can you see which one it is?