Alan Quartermain has fallen off the side of a mountain while adventuring in Sout

Question

Alan Quartermain has fallen off the side of a mountain while adventuring in South America. He has stopped his fall by grabbing a tree root, but just barely. If his footing slips and he takes one more step backwards, he will fall to his death. He must take 3 steps up the very crumbly face of the cliff from his current spot to reach the top and safety. Each time he tries to take a step, the odds are 2 in 3 that he moves up the cliff towards safety, and 1 in 3 that he moves down the cliff towards his death. What are Alan’s chances of survival from his dire predicament?

Explanation / Answer

Probability of moving up the cliff towards safety when he makes one step = 2/3

To survive, he need to make three steps without falling.

Probability of him making three steps without failing and hence, saving himself = 2/3 x 2/3 x 2/3

= 8/27

= 0.296

Alan's chance of survival from his dire predicament = 8 in 27

= 0.296

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