Poor Frog
There is a frog stuck in a 10 feet deep well. Every day in the morning he jumps up 3 feet, and at night slides down 1. Each night there is a 50% chance that there is a 50% chance of rain, and a 50% chance that there is no chance of rain. When it rains, the poor frog slides down an additional 1 feet. What is the probability that the frog would escape in 5 days?
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1 solution
There is a 50% chance that there will be a 50% chance of rain, so there will be a 25% chance of rain every day. The frog goes up a 2 feet every day, since he goes up 3 then slides back down 1. We can see that he can escape in 5 days only if he is capable of jumping up a total of 2 feet per day. This means that for all 5 days it must not rain so the frog can escape. The chance of rain, again, is 1/4. Since we need it to not rain for 5 days, the probability would be (3/4)^5, which is equal to 243/1024.