given that a bag contains 4 balls and 2 balls are drawen at random and both found to be white

Since you said nothing about what are the colours of the 4 balls and the 2 balls, nothing can be calculated from here.

I know you must have the question with you, so, the probability of having a white ball is the number of white balls over the total number of balls in the bag.

For two white balls, you have to multiply the probability of having one ball, then another white ball, which is given by the number of white balls remaining over the number of balls remaining in the bag.

Consider all the possible outcomes:
2 of 4 balls are white
3 of 4 balls are white
or all balls are white

Each has an equal chance of being true, therefore each has a probability of 1/3.

The probability of all the balls being white is 1/3.

No the question is not. There is an improper use of the connectives (punctuation and grammar at the same time) which render the question vague.

What if the actual question was:
"Given that a bag contains 4 blue balls and 2 white balls. Two balls are drawn at random. Find the probability that both are found to be white."

Then, your answer won't be correct.

If the actual question was:
"Given that a bag contains 4 balls. Two balls are drawn at random and are found to be both white. Find the probability that all balls in the bag are white."

Then, that is the situation you are saying.
2 of 4 balls are white = 2C2 = 1 way
3 of 4 balls are white = 3C2 = 3 ways
4 of 4 balls are white = 4C2 = 6 ways

So, probability that all the balls were white from the start is 6/10 = 3/5

Your answer is wrong again.

I have to agree with Unk that the problem can't be solved given the information provided. The answer depends on the probability of a ball being white in the first place, which we don't know. Perhaps these two examples will make this clear:

1. Suppose the 4 balls in the bag were selected from a group of 2 white balls and 8 red ones. Clearly the probability that all four balls in the bag are white is 0, even though we happened to pull out the two white ones with our first two draws.

2. Suppose the four balls in the bag were selected from a population of 10 white balls. Then the probability of all 4 balls being white is 1.

So to answer the OP's question -we need to know something about the population of white balls within the larger set from which the 4 balls in the bag were selected.