That is, the mean of all coin flips approaches zero as the number of flips increases. This follows by the finite additivity property of expectation:  A similar calculation, using the independence of the random variables and the fact that formula_12, shows that:  This hints that formula_14, the expected translation distance after "n" steps, should be of the order of formula_15. In fact,  This result shows that diffusion is ineffective for mixing because of the way the square root behaves for large formula_17. How many times will a random walk cross a boundary line if permitted to continue walking forever? A simple random walk on formula_1 will cross every point an infinite number of times. This result has many names: the "level-crossing phenomenon", "recurrence" or the "gambler's ruin". The reason for the last name is as follows: a gambler with a finite amount of money will eventually lose when playing "a fair game" against a bank with an infinite amount of money. The gambler's money will perform a random walk, and it will reach zero at some point, and the game will be over. If "a" and "b" are positive integers, then the expected number of steps until a one-dimensional simple random walk starting at 0 first hits "b" or −"a" is "ab". The probability that this walk will hit "b" before −"a" is formula_19, which can be derived from the fact that simple random walk is a martingale. Some of the results mentioned above can be derived from properties of Pascal's triangle. The number of different walks of "n" steps where each step is +1 or −1 is 2"n". For the simple random walk, each of these walks is equally likely. In order for "Sn" to be equal to a number "k" it is necessary and sufficient that the number of +1 in the walk exceeds those of −1 by "k". It follows +1 must appear ("n" + "k")/2 times among "n" steps of a walk, hence the number of walks which satisfy formula_20 equals the number of ways of choosing ("n" + "k")/2 elements from an "n" element set, denoted formula_21. For this to have meaning, it is necessary that "n" + "k" be an even number, which implies "n" and "k" are either both even or both odd.