Randomly vs Sequentially selecting an item out of a box? Say you have $n$ envelopes in a box placed in some order, and one of them is a special envelope. On average, would you have a higher, lower or equal chance of picking the special envelope if you went in sequential order (left to right) versus if you randomly chose an envelope instead (assuming without replacement)? (Or without replacement)
The two probabilities are the same for all values of $n$.

Assuming that the order of the envelopes was determined randomly (ie they were thoroughly shuffled), then drawing sequentially from this randomly-ordered pack is the same as dealing cards from the top of a shuffled deck. This is the same as drawing from random positions without replacement from a shuffled deck. The double randomness in the second method (random shuffle followed by a random draw) does not make the order in which the cards are drawn any more random. A single random shuffle is as good as a thousand.

In both cases each envelope is drawn only once, and the order in which they are drawn is decided at random. After each draw there are the same number of undrawn envelopes remaining. The special envelope is at a random position among the remainder, so it has an equal chance of being at any position among the remaining cards.
as n goes towards infinity, the two probabilities , namely the one you pick randomly and the one you pick in an order will tend towards the same value. 

Why? Think about it as a deck of cards, say I want the hearts 6, I first pick a card and pray that it is hearts 6, it is not and I pick again and I continue until I get hearts 6, this chance on the first try might be (1/52), but if I go in an order I would get (1/52)^n.