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Independence of the judiciary means that judges can make decisions without fear of losing their jobs based on the decision. That's the theory on why judges are given lifetime appointments. 

For example, the Supreme Court could vote to overturn Roe v. Wade if a case came before them, and they wouldn't have to worry about being fired or re-elected based on how they came down on the decision.
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Independent judiciary means the judicial institution that acts independently and is not answerable to the other institution such as the legislature and
 executive. this supreme and independent power is given to the judiciary because it governs the country and ensures law and order in the country.
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