Long Legged Legolas

Probability Level 3

Long Legged Legolas decides to climb up a flight of 200 stairs. He starts from the floor, and can go up 1-6 steps at a time. (He rolls a standard 6 sided die each time to decide which one). What is the probability that at some point he will land on the 100th step?

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The answer is 0.286.

1 solution

Geoff Pilling
Mar 20, 2019

On average, he will go up 3.5 stairs for every step.

So, for large enough $n$ , like 100, the probability of the $n$ th step being stepped on is $\frac{1}{3.5} \approx 0.286$

What if he is on 99th step and gets 6 (or any other number >1) on his dice?

Mr. India - 2 years, 2 months ago

If he is on the 99th step then there would be a $1/6$ chance of stopping on the 100th step, but it's not a certainty that he will land on the 99th step. He could land on the 98th step and roll a 2 (thus skipping the 99th step), or land on the 97th step and roll a 3, etc.. So the probability calculation has to involve all these possibilities. Geoff's solution is a clever short-cut, but a more formal analytic solution involves recursive equations.

Brian Charlesworth - 2 years, 2 months ago

Is 1/3.5 also approx as n is large? What will be the solution you said?

Also, there are 100 steps only in the staircase, so isn't it a must for him to get on 100th step?

Mr. India - 2 years, 2 months ago

@Mr. India – In the limit as $n \to \infty$ the probability goes to $2/7$ , but $n = 100$ is large enough that the probability is $2/7$ for the 100th step to more than three decimal places. As for your last question, that's a good point; perhaps Geoff should have made the full staircase, say, 1000 steps instead so that LLL could walk over the 100th step. @Geoff Pilling What do you think about this edit suggestion?

Brian Charlesworth - 2 years, 2 months ago

@Mr. India – To that point, I've updated the total number to be 200. Thanks! (The answer is still the same)

Geoff Pilling - 2 years, 2 months ago

Long Legged Legolas

The answer is 0.286.

1 solution

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