At a local high school, the probability that a student is absent on any given day is about 0.08. however, the probability that a student is absent given it is friday is 0.20. are these two events independent?

No.

Step-by-step explanation:

Your keyword here is "given." If we calculate the probability of an event, GIVEN that one event occurs, too, then we have dependent events, because the outcome of one event relies on the other. For this scenario, you have the probability of absence on any day of the week is 0.08, but then they say "well, but if it's Friday, that probability actually goes up quite a bit." This tells you that your events aren't independent because the day being Friday increases the chances of a student being absent from the baseline 0.08 probability.

not only in mathematics, scientific proof also are required in the other field of science. also maths proofs are of particular structure, containing steps and logics. so proving is like doing science, it is science. that's why it is important to prove. 

an inaccurate proof leads to errors, or stating results that are not accurate.  which wil mess the probelm up you dont want this