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The DMG description of Potion of Longevity says:

When you drink this potion, your physical age is reduced by 1d6 + 6 years, to a minimum of 13 years. Each time you subsequently drink a potion of longevity there is a 10 percent cumulative chance that you instead age by 1d6 + 6 years.

What does a 10% cumulative chance mean in this context?

Assuming that it means the chance increases by 10% each time, I find the statement awkward because usually when you state a cumulative probability, you are stating the total, not the amount that is added. That is, if we are to add 10% each time, then after 3 uses, there's actually a 20% cumulative probability that it ages you (not a 10% cumulative probability).

By a strict reading, I'd read that the probability is 10% each time after the first, but that is almost certainly not what was intended.

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Cumulative chance does mean that the chance accumulates each time, yes. Each time you drink it after the first, there is a chance that it will age you instead of youth-en you. That chance is 10% initially, but increases by 10% each time.

So the first time you drink it, it has its normal effect. The second time, there's a 10% chance it has the opposite effect and ages you. The third time, a 20% chance of aging. The next time, a 30% chance; the next 40%. And so on.

This chance doesn't ever reset, it just accumulates.

(This does seem to be a pretty poor potion of longevity. The local Better Magical Business Bureau will probably need to open an investigation into deceptive advertising and naming practices.)

As for the wording, this isn't actually incorrect, perhaps just unfamiliar. A "cumulative chance" can be used as part of a statement of an observed or imposed rule or physical law, or it can be used as part of a sentence describing the current state of a cumulative chance. The normal usage is the former, with the latter being derivative of the former and more often seen in summaries or after-the-fact status reports and similar contexts. The default meaning of "cumulative chance" is describing how the chance accumulates, and is what should be assumed if in doubt.

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