What are independent trials, and how do they relate to the definition of probability?

In Probability we use the term Bernoullian trials for independent trials.

A sequence of trials are said to be Bernoullian if (i) there are only two outcomes for each trial ( say a success or a failure) (ii) Consecutive trials in the sequence are independent and (iii) probability of success in every trial is a constant (namely p) It is clear that no trial depends on the previous one here. Consider drawing of two cards consecutively from a well shuffled deck of cards. Consider the event that cards are both diamonds. If the draw is with replacement the trials are independent ( Bernoullian) . Otherwise they are dependent. In the former case the probability is ( 1/4) x (1/4) In the later it is( 1/4) x (12/51)