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# Probability

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While staying in a 15-story hotel, Polya plays the following game. She enters an elevator on the $$6^{\mathrm{th}}$$ floor. She flips a fair coin five times to determine her next five stops. Each time she flips heads, she goes up one floor. Each time she flips tails, she goes down one floor. What is the probability that each of her next five stops is on the $$7^{\mathrm{th}}$$ floor or higher? Express your answer as a common fraction.

Is there any way to split it into cases or use complementary probability? I don't know how to approach this problem... any help would be appreciated!

Aug 26, 2018

#1
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I don't really have much time right now, but basically I used casework counting, like if Polya goes up or down, then make subcases. My final answer was $$\frac{3}{16}$$. Since I quickly did this, my answer might be incorrect. Hopefully I helped!

- Daisy

Aug 27, 2018
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I agree with Daisy's result.

In the following let u = up, d = down and x - either up or down.

The only possible ways in which Polya stays on floor 7 or above at every stop are as follows (where moves are ordered from left to right):

u u u x x   p1 = (1/2)3

u u d u u   p2 = (1/2)5

u u d u d   p3 = (1/2)5

p1+p2+p3 = 3/16

.

Aug 27, 2018
edited by Alan  Aug 27, 2018
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Thanks for your solutions! They really helped!

Aug 27, 2018