It's not a bell curve, the probability density function of an exponential distribution looks like this:
Note that the average block time is 10 minutes (1/λ), but 50% of the blocks are less than 6.93 minutes (the median ln(2)/λ).
The reason that it makes no difference is: whether the geometric distribution (which counts individual hashes) is calculated for difficulty 1 or difficulty 10000000, the probability of a quantile (such as calculating the probability of a block taking 100 minutes) is the same within several decimal points. In fact it is already identical to the exponential (continuous) function with three digits by difficulty 1, higher difficulty just makes the density function of geometric converge to exponential with even more digits of identity.
http://math.stackexchange.com/questions/93098/how-does-a-geometric-distribution-converge-to-an-exponential-distribution
Here's a monte carlo simulation of a geometric distribution density, the red line is exponential:
See how much like the exponential it is? This example shows a 0.1 probability; Bitcoin's current probability is 0.0000000000000000231. With a lower probability,the steps disappear and the distribution converges to exponential.
So the answer is: the probability of a block taking 62 minutes (given a constant hashrate equivalent to difficulty) is 0.2029%, whether the difficulty is 1 or a billion.
Note that the average block time is 10 minutes (1/λ), but 50% of the blocks are less than 6.93 minutes (the median ln(2)/λ).
The reason that it makes no difference is: whether the geometric distribution (which counts individual hashes) is calculated for difficulty 1 or difficulty 10000000, the probability of a quantile (such as calculating the probability of a block taking 100 minutes) is the same within several decimal points. In fact it is already identical to the exponential (continuous) function with three digits by difficulty 1, higher difficulty just makes the density function of geometric converge to exponential with even more digits of identity.
http://math.stackexchange.com/questions/93098/how-does-a-geometric-distribution-converge-to-an-exponential-distribution
Here's a monte carlo simulation of a geometric distribution density, the red line is exponential:
See how much like the exponential it is? This example shows a 0.1 probability; Bitcoin's current probability is 0.0000000000000000231. With a lower probability,the steps disappear and the distribution converges to exponential.
So the answer is: the probability of a block taking 62 minutes (given a constant hashrate equivalent to difficulty) is 0.2029%, whether the difficulty is 1 or a billion.