Statistics and probability - Independent Events and Product Rule #statistics

Statistics and probability: Independent Events and Product Rule

The notion of Independent Events plays a vital role in material virtually in all areas of applied statistics. In the die-tossing experiment, we note that P(B|A) = 2/5 whereas P(B) = 1/3. That is, P(B|A) is not equal to P(B), indicating that B depends on A.
Now consider an experiment in which 2 cards are drawn in succession from an ordinary deck, with replacement. The events are defined as
A: the first card is an ace,
B: the second card is a spade.
Since the first card is replaced, our sample space for both the first and the second draw consists of 52 cards, containing 4 aces and 13 spades. Hence,
That is, P(B|A) = P(B). When this is true, the events A and B are said to be independent. In other words, the occurrence of B had no impact on the odds of occurrence of A. Here the occurrence of A is independent of the occurrence of B.
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