A) Define independent event. If A and B are independent event then show that thefollowing pair is also independent:i.A andBii.A andB
Question
A) Define independent event. If A and B are independent event then show that thefollowing pair is also independent:i.A andBii.A andB
Solution
A) An independent event in probability theory and statistics is an event that is not affected by other events. In other words, the probability of the event occurring remains the same regardless of whether the other events occur or not.
If A and B are independent events, this means that the occurrence of A does not affect the occurrence of B, and vice versa. Mathematically, this is expressed as P(A ∩ B) = P(A)P(B).
i. A and B: Since A and B are given as independent, their independence is already established.
ii. A and B: This seems to be a typo in your question as it is identical to the first pair. If you meant A and not-B (the complement of B), then they are also independent. This is because the occurrence of A does not affect the occurrence of not-B, and vice versa. Mathematically, this is expressed as P(A ∩ not-B) = P(A)P(not-B). Since A and B are independent, P(B) = 1 - P(not-B), and we can substitute this into the equation to verify the independence of A and not-B.
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