It's not a uniform distribution after the first measurement, t_obs. That enables us to update the distribution, and it becomes a decaying one.
I think you mistakenly believe the distribution is still uniform after that measurement.
The best guess, that it will last for as long as it already survived for, is actually the "median" of that distribution. The median isn't the highest point on the probability curve, but the point where half the area under the curve is before it, and half the area under the curve is after it.
It's not a uniform distribution after the first measurement, t_obs. That enables us to update the distribution, and it becomes a decaying one.
I think you mistakenly believe the distribution is still uniform after that measurement.
The best guess, that it will last for as long as it already survived for, is actually the "median" of that distribution. The median isn't the highest point on the probability curve, but the point where half the area under the curve is before it, and half the area under the curve is after it.
And the above equation is consistent with that.