Probability that at least one of two students will solve a random problem 
Question
Ashmit can solve 80% of problem given in a book and Amisha can solve
  70%. What is the probability that atleast one of them will solve a
  problem selected at random from the book? 
Answer
there is a 20% chance that Ashmit will NOT solve. 
  Of those 20%, Amisha
  has a 70% of solving them. 20*70% = 14% Add 80 + 14 = 94%
If you do Amisha 70% first, you get the same answer.

This question and answer is taken from https://in.answers.yahoo.com/question/index?qid=20130109042412AAg1fFa
Many sites give the same answer. But, I do not think this answer can be right.To me, we can find answer only if we know how many problems can be solved by both of them.
For example, suppose total questions are 100.
80 questions are solved by Ashmit correctly
70 questions are solved by Amisha correctly 
say 60 questions are solved by Ashmit and Amisha  correctly.
then, required probability will be 80+70-60=90

So it is important how many questions are correct by both persons.
Where am I going wrong in my approach?
 A: The model answers seem to assume that Ashmit and Amisha's abilities to answer particular questions are independent, so 


*

*probability both can solve $= 0.8 \times 0.7= 0.56$

*probability Ashmit can solve but Amisha cannot $= 0.8 \times 0.3 = 0.24$

*probability Ashmit cannot  solve but Amisha can $= 0.2 \times 0.7 = 0.14$

*probability neither can solve $= 0.2 \times 0.3= 0.06$


with $0.56+0.24+0.14 = 0.94$.
You are correct that there are other possibilities without independence,  with extremes such as 


*

*probability both can solve $= 0.7$

*probability Ashmit can solve but Amisha cannot $= 0.1$

*probability Ashmit cannot  solve but Amisha can $= 0$

*probability neither can solve $= 0.2$


with $0.7+0.1+0 = 0.8$, or in the other direction  


*

*probability both can solve $= 0.5$

*probability Ashmit can solve but Amisha cannot $= 0.3$

*probability Ashmit cannot  solve but Amisha can $= 0.2$

*probability neither can solve $=  0$


with $0.5+0.3+0.2 = 1$.
A: You are adding more information to the problem, thus changing the probability.
Compare: Rolling a dice you have $1/6$ to get a $5$. However if I add the information that I did not roll a $1$, then the probability of having rolled a $5$ becomes $1/5$.
In the same way if you add information about how many problems they both can solve correctly, then you change the probability, because we have given more information. That is we are restricting the problem to a certain case where, for instance, both together can solve 60 questions, not to the general case where this number may be anything.
The extreme case of this is ofcourse if there are 100 questions and they together may solve exactly 50 questions. As one of them solve $70$ and the other solve 80 problems, each problem is solvable by someone of them. Thus the probability becomes $1$ of one of them being able to solve a certain problem.
A: The problem is that the original problem does not specify whether Ashmit's and Amisha's problem solving capabilities are independent of each other.
In the answers you cite, it is assumed that their problem solving capabilities are independent.
In your approach, their problem solving capabilities are not independent, which gives a different outcome.  
In particular, if Amisha can only solve problems which Ashmit can solve, then there is a $80\%$ change that a random problem will be solved.
If in the other extreme case Amisha can solve all the problems Ashmit cannot solve and some problems Ashmit can solve, then there is a $100\%$ chance that a random problem will be solved.
