# Calculating average cost of water including possible lottery winnings

There is a 1.5 liter water bottle for 72 cents in the shop. 10 cents is the price of the plastic bottle that is retrievable after you return the bottle, and 62 cents is the price for the liquid itself. Price for the liter of water would be 41.3 cents, but there is a catch: every bottle has a lottery and gives 1/5 chance to win another 0.5 liters water bottle for free (liquid and the 10 cent plastic bottle). The question is how much one liter of liquid actually cost on average, including the chances of winning extra bottles.

This means on worst case scenario you can buy one water bottle for 72 cents, return the bottle for 10 cent and the final one liter price will be 41.3 cent. Best case scenario would be that after buying one bottle, you win another bottle (that means you get more water [0.5l] and bottle that can be retrieved for 10 cent), that actually wins another 0.5l bottle and so on, so one liter of water would cost nothing, or even negative amount, however this is highly unlikely and happens very rarely.

I've already tried to calculate this by simulating buying with computer program, and the answer from a lot of tries is about 37,84 cents per liter, but I think this can be solved with infinite series algorithms.

TLDR: All plastic bottles are worth 10 cents. First 1.5l bottle is bought for 72cents including the plastic bottle. It has 1/5 chance to win another 0.5l bottle for free, that can also win another bottle recursively. How much on average 1l of liquid cost.

Thank you

HINT

$1.5\text{ liter} + \frac15\text {liter} + \frac1{25} \text {liter} + \ldots = L$

$L$ is expected number of liters you get after a purchase.

$62\text{ cents} - \frac15\cdot10\text{ cents} - \frac1{25}\cdot10\text{ cents} - \ldots = C$

$C$ is expected cost of a purchase after all bottles are returned for their deposit. If all bottles are not to be returned, the equation will need to be modified accordingly.

Cost per liter is what in terms of $L$ and $C$? How do you calculate sum of geometric series?

• The total L is 1.625 and the C equals to 59.5 cents. The answer is 36 + 8/13. I just wonder why answer differs by that much when simulating purchases programmatically – Augustinas Aug 31 '17 at 21:48
• Yes, that's a good question. I wondered about that as well. Did the simulation assume all bottles would be returned to get deposit back? More generally one would need to understand that the calculations here used (or didn't use!) the same assumptions as the simulation. Maybe you'd like to analyze the simulation and add that to the OP, or alternatively, link the OP to the source code. – Χpẘ Sep 1 '17 at 19:52