Found a probability problem on twitter and found the solution to be counterintuitive:

At 1 minute to midnight, 10 apples fall into a sack. The same happens at half a minute to midnight, then at a quarter minute to midnight, and so on. At each such event you remove an apple randomly from the ones still present in the sack.

What is the probability that at midnight strike the sack will be empty?


For the first iteration of apples: 10 falls, 1 gets immediately removed leaving 9 in the bag.

Second iteration of apples: 10 falls, 1 gets immediately removed leaving 19 in the bag.

The bag appears to approach infinitely larger size, so it would never be empty.

What am I thinking about incorrectly?

  • $\begingroup$ I agree. The way the problem is stated, the apples would grow. In fact, there is nothing random about the number of apples left after the $n$th stage. However, the question "how many left at midnight" is ambiguous. One way is to define it is as a limit, in which case it would be infinite. $\endgroup$ – TorsionSquid Oct 24 '17 at 13:32
  • $\begingroup$ This is some well-known "trick" question to show the problems with the concept of infinity when applied careless. While it seems the number of apples approaches infinity, the probability for each individual apple to be still contained in the sack at midnight is zero. $\endgroup$ – M. Winter Oct 24 '17 at 13:38
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    $\begingroup$ I think the problem is more "paradoxical" without randomness. We picture the sack as a stack, and we always remove the apple from the bottom of the stack. In this way, every apple put in will eventually get taken out. $\endgroup$ – TorsionSquid Oct 24 '17 at 13:41
  • $\begingroup$ @M. Winter What would fix the infinity problem in this puzzle? $\endgroup$ – AltoidsBenefitsH Oct 24 '17 at 14:00
  • $\begingroup$ @AltoidsBenefitsH As you are talking about an infinite process, you should start by defining the kind of limit you are considering. I admit that I never saw above "riddle" in a rigorously stated form. $\endgroup$ – M. Winter Oct 24 '17 at 14:24

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