What is the probability that a glass of water contains a water molecule that you’ve drunk before?

Surprisingly, it’s around 100%.

Suppose you've drunk 2 litres of water every day for 30 years. That's roughly: 2×365×30=~2 × 10⁴ kg of water.

I will take the Earth’s radius as 10,000km and assume 60% of the Earth’s surface is covered by water. The hard part is estimating the average depth of all this water, which I’ll tentatively guess is 5000m.

That means there are 4π (10,000,000)² x 0.6 × 5000 × 1000 =~ 4 × 10²¹ kg³ of water on Earth total.

So the probability that you have drunk any molecule of water is (2 × 10⁴)/( 4 × 10²¹) = ~5 × 10⁻¹⁷

Take a 300 mL glass of water with a mass of 0.3 kg. If you do any chemistry, you’ll know that the molar mass of water is 0.018 kg/mol, but let’s round this to 0.02kg/mol.

So, a glass of water contains ( 6 × 10²³)*(0.3)/(0.02)=~9× 10²⁴ molecules

Therefore, in any glass of water, there are about (9× 10²⁴)x(5 × 10⁻¹⁷)=4.5×10^8 water molecules that you have drunk before.

The probability that there ISN’T 1 or more molecules you have drunk before is about 1−e^(-4.5×10^8)=~1

Thus, the probability that there is a water molecule you have drunk before is essentially 100%.