Gambler's Fallacy again, looking for a succinct explanation of a variant of Martingale

[silentdiver]

A friend and I have been discussing an approach, and I'm looking for the most clear and succinct way to refute it. We agreed that I'd put it to the wise folk here, so here goes.

I'm happy to be shot down in flames; that's precisely what I'm after, but ideally expressed in an irrefutable and convincing form that I can pass on.

The essence of it is, to take a roulette analogy; waiting for 10 blacks, then betting on red and increasing the stake on each subsequent black until there's a red.

The theory being that since there have already been 10 blacks... that... and I feel silly saying it... but that the next red is 'due' 10 spins earlier than if we'd not waited for the 10 blacks first.

I'm having trouble expressing why this is no different to not waiting until there are 10 blacks in a row, and just betting in exactly the same way, I know it's probability-istically (!) identical, and so pointless pursuing (and a sure loser ultimately); but I can't convince my friend why.

The actual context is horse racing, and a Martingale staking plan; waiting until X losses in a row, then starting to increase the stake until there's a win - the same as the roulette analogy.

The counter-argument put to me is that since the losses 'started' 10 bets earlier (bets that weren't placed) that the losses from those 10 not-placed bets won't be lost and the win will come sooner - but of course, since the bets weren't placed, then there aren't 10 losses already, we're not at 'bet #11' in a losing streak, we're at bet #1.

Much more to the point, we are always at bet #1 and always will be, regardless of how many wins or losses have happened or whether we bet on them or not.

The whole '10 losses in a row' thing is purely an intellectual/statistical construct that has been superimposed on 10 unconnected events in the past; it is of no use in predicting anything.

Can anyone more eloquent than I come up with a way to make that crystal clear; specifically squashing the counter-argument ?

I've looked at several YouTube videos on the fallacy, but not found one that deals specifically with the 'Wait for X losses first' idea.

The best i can come up with is along the lines of "The horse has no idea how many bets we recently placed on other horses at other tracks around the world - so how can the bets that were or weren't placed have any effect on whether this one wins or not" but that doesn't do the job, nor do paraphrases of any of the above.

Thanks