8

A frog is trapped in a well, just 1 meter below the lip. On sunny days, the well is dry, and the frog is able to climb up 1 meter. On rainy days, the well is wet and the frog slides down 1 meter. If it is rainy 2 out of every 3 days and the well is infinitely deep, what are the frog's chances of ever reaching the lip and escaping?

H Rogers
  • 692
  • 4
  • 9

1 Answers1

9

This is an example of the

Gambler's ruin problem. Here, we consider "escaping the well" as the equivalent of becoming bankrupt, and "never escaping" as the equivalent of becoming infinitely rich.

Using that,

The probability is $p = \frac{2}{3}$ and starting position $i=1$. We plug this in the formula of $1 - \Big(\frac{1-p}{p}\Big)^i = 1 - \frac{\frac{1}{3}}{\frac{2}{3}} = \boxed{\frac{1}{2}}$.

sedrick
  • 2,006
  • 10
  • 15
  • 2
    Haven't seen this version of the problem before, nice! To me it's a neat paradox that the frog has a chance of never escaping, but will never have no chance of escaping. – H Rogers May 20 '20 at 17:56
  • 4
    I just spent about 30 minutes writing out the P(x) equations for each meter and doing a convergence sum. I was about to post my answer and then I saw this and how simple it was lol – Ankit May 20 '20 at 18:02
  • 2
    @Ankit, The solution you mentioned is included in the proof, so you are doing the right way. It is just that it already has a name (I was not familiar with this name before, too) – justhalf May 21 '20 at 06:14
  • 1
    @justhalf Lol yeah I got 1/2 as well, I'm just saying that I spent so much time on a problem that takes 2 minutes lol. – Ankit May 21 '20 at 19:04
  • I'm still having trouble wrapping my head around this. How could it be that there's a 50% chance the frog won't escape when the only way the situation ends is when the frog escapes? The situation should theoretically go on forever until the frog escapes, why is there ever a chance the frog won't escape? – Cotton Headed Ninnymuggins Jun 27 '20 at 17:02
  • @CottonHeadedNinnymuggins Yes, it may seem a bit paradoxical, but if the probability of something happening is large enough, the long-term probabilities of events can approach zero. You can imagine this situation: Suppose you are gambling and each round you spend a dollar. Now suppose you start with 100, and suppose you have a 99.99% chance of winning 1,000,000 in a game, and 0.01% chance of losing the 1 dollar you spent. If you play infinitely, sure, there's always "a chance" of going broke, but the limit of that event happening approaches zero as you play forever. – sedrick Jun 28 '20 at 16:10
  • @sedrick Does that mean the probability that the frog makes it out *approaches* 50%? – Cotton Headed Ninnymuggins Jun 28 '20 at 19:43