Keeping Advantage Toggle on and using the left number is fair?

First up I'm not a mathematician or a programmer but this got me thinking and if I'm wrong about this I'll be happy to be told so. So I've noticed some DMs just keep the Advantage Roll Toggle on and use the left number when the results come up. My question is, isn't the software technically rolling twice thus doubling the chances of a result? Statistically if you rolled 1000 attacks with Advantage Off and 1000 attacks with Advantage On but used the left result would the number of successes be roughly the same? It feels like to me they hit more often than in games I DM where I only turn Advantage on when needed but that could be just random chance and perception. Just curious.

Think of it like rolling a blue d20 and a red d20 every time at a physical table but only ever reading/using the blue result. It's the same principle, you might suffer the sense of loss at the occasional red natural 20 but overall it doesn't affect the outcome of the blue die. In general, I find it speeds up play significantly which is a huge bonus worth any potential cost (not that there is one...)

What you're talking about is called Gambler's Fallacy . It's the mistaken belief that two otherwise unrelated random events will influence each other. If you roll a 20 sided die, the chance of a 20 is 1 in 20, 5%. It doesn't matter how many times you roll it, it doesn't matter if the previous roll was a 20.  Because each face has an equal probability, there should be a statistically even distribution of rolls, given a large enough sample size , but nothing in the universe or Roll20 enforces that. Statistics are why people play the odds abs why casinos are rich. 

Ziechael said: Think of it like rolling a blue d20 and a red d20 every time at a physical table but only ever reading/using the blue result. It's the same principle, you might suffer the sense of loss at the occasional red natural 20 but overall it doesn't affect the outcome of the blue die. In general, I find it speeds up play significantly which is a huge bonus worth any potential cost (not that there is one...) Thanks for clarifying that Ziechael

The Aaron said: What you're talking about is called Gambler's Fallacy . It's the mistaken belief that two otherwise unrelated random events will influence each other. If you roll a 20 sided die, the chance of a 20 is 1 in 20, 5%. It doesn't matter how many times you roll it, it doesn't matter if the previous roll was a 20.  Because each face has an equal probability, there should be a statistically even distribution of rolls, given a large enough sample size , but nothing in the universe or Roll20 enforces that. Statistics are why people play the odds abs why casinos are rich.  Thanks Aaron. Probably why I'm not a gambler.

One subtle point, though, is for the 5e Hobbit's Lucky Trait, it would be inappropriate to use the second die roll if the first comes up as a 1, since the second die roll is at that point known. 

Bill (Do Not Kill) said: One subtle point, though, is for the 5e Hobbit's Lucky Trait, it would be inappropriate to use the second die roll if the first comes up as a 1, since the second die roll is at that point known.  Only if you can envision a situation in which you wouldn't re-roll a result of 1. There's no cost to it, and if the second roll is a 1, you can just take it as read that you re-rolled a physical die and came up with another 1.

It's the same with inspiration, in those events I ask my players to roll again and take the first roll in that one too. For me the 2nd roll is only ever for advantage and disadvantage, nothing else.