Not quite. The expected number of sequences until you
exactly 5 losses in a row is 41772.04. The expected number of loss runs until you hit
more than 5 losses in a row 3e05.
Keep in mind that the variance is large. If the expected number of loss runs is 1/3e05, then 95 times out of a hundred that you experiment, you'll hit a hit loss streak of 6 or more at between 7,595 and 1,106,661 sequences.
So, a loss run is any martingale sequence? Not including just 1 loss. That means at least 2 losses in a row? Or does loss run mean any sequence of losses, including a single loss?
"Losses in a row" can be any positive integer from zero up: 0, 1, 2, 3 .... For example loss, loss, win, loss, win, win is 2 losses in a row, followed by a one loss sequences, followed by a zero loss sequence.
So "Losses in a row" is basically how many wins, because that includes zero loss sequences, except if there are "Wins in a row" which are counted as one up to the next loss. Correct? This translates to approximately 87.7779% of the number of rolls? Or the other way around, about 13% of the number of rolls is the losses, therefore, the "Losses in a row" sequences.
Also, 95 times out of a hundred, I will hit a loss streak of 6 or more after 7,595 sequences, but before 1.2 million sequences. Which means I'm almost guaranteed to hit the loss streak after a million rolls.
I think the debate now is whether just-dice is really memoryless or not, but that's another issue. We can all agree that it might indeed be actually memoryless. Or maybe that's just Gambler's Fallacy whispering in my ear again ...
Dung beetles navigate using the milky way galaxy ... (okay, just having some fun here.)