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If a student copies his assignments from his friend he would get 80 marks. If he had done the assignments independently he would have scored 50 marks out of 100 and if the teacher finds he is cheating he will be penalized and will be given 0 marks. The probability that the teacher catches his cheating is 0.3. If he copies 10 such assignments, what is the probability that he will lose more marks with copying than by doing his independent work independently?

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I'm a tad bit unsure,but still my approach would be as below:- :)

The experiment is conducted 10 times,and we need to consider the cases and add up those cases in which he gets less than (50*10=500,if he gives the exam fairly),and maximum before 500  is when he cheats and remains uncaught for 6 times out of 10,which makes his score (80*6=480).

Now,let's assume p be the probability he cheats and succeeds in that,(p=1-0.3=0.7) ,and probability of failure be (q=0.3),now it's just adding up the binomial terms for which number of success<=6,I'll create a table below:-

Success Marks Probability
6 80*6=480 $\binom{10}{6}(0.7)^{6}0.3^{4}$
5 80*5=400 $\binom{10}{5}(0.7)^{5}0.3^{5}$
4 80*4=320 $\binom{10}{4}(0.7)^{4}0.3^{6}$
3 80*3=240 $\binom{10}{3}(0.7)^{3}0.3^{7}$
2 80*2=160 $\binom{10}{2}(0.7)^{2}0.3^{8}$
1 80*1=80 $\binom{10}{1}(0.7)^{1}0.3^{9}$
0 80*0=0 $\binom{10}{0}(0.7)^{0}0.3^{10}$

Summing up the above binomial probability terms will yield the ans

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