Question

why do we see water droplets on the outer surface of a glass tumbler containing crushed ice

Answer 1

The vapour outside the glass will start to condense because of the ice inside it. Leaving water droplets on the surface.

Answer 2

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We usually see water droplets on the outer surface of a glass tumbler containing crushed ice, because the coming in contact with the cool ice or that cool atmosphere get condensed and turn their state from gas to liquid ( water).