Can some one explain what is the difference between rolling three dice together and rolling a single die three times? What is the probability that the sum equals 4 for both cases?
Arguably, rolling a single die three times gives more information than rolling three dice simultaneously, as in the first case, we can definitely identify the order of the rolls. (E.g., we can distinguish between, say, 1,4,3 and 4,3,1.) The advantage of considering the order of the rolls to be important is that then you have an equiprobable sample space--all $216$ outcomes are equally likely.
In probability event(theoretically), we usually assume ideal condition.
The probability of a die with result from 1 to 6 is 1/6 regardless what happens to other dies either thrown at the same time or thrown few seconds later. We assume each event of throwing a die is independent, therefore each event is also independent of 'time' each die is thrown.
There will be time difference.
Throwing all three dies at time 00:00:00. vs. Throwing first die at 00:00:00, second at 00:00:01, third at 00:00:02.
But again in the ideal condition, the outcome of a die is not dependent on the time it is thrown unless there're some kind of events that will effect the outcome of a die at a specific time.
So basically two events (throwing three dies at the same time or throwing three dies at different times) will have same probability for any probability events (e.g sum of three dies = 4, multiplication of three dies = 30, and so on)
Assuming 6 sided dice, there are $6^3$ ways the dice can land. To make a 4 can only be done in three ways (1,1,2; 1,2,1; 2,1,1). So the probably is $3/216=1/72$.
Now, if you're allowed to stop after one or two throws, then that changes it a bit. There's a 1/6 probability on the first throw, 1/12 probability on the second throw (1,3; 2,2; 3,1), and 1/72 on the third, so you'd have to add those together.