Geometric distributions.

Dec 2015
25
1
Worcester, UK
Question I am struggling on is attached.

I can do all the questions bar the last one; apparently, P(X=3) (where X has a geometric distribution) is incorrect. I can't really see why this is as P(X=3) is defined as 'the probability of failing two times and then having a success on the third trial." Why isn't this the answer?

Thanks.
 

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romsek

MHF Helper
Nov 2013
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2,993
California
There are two levels of probability going on in the last problem.

First we have to compute on a given day the probability of lighting the fire in under 4 attempts.

This probability generates a new distribution for the probability of being able to light the fire in 4 attempts on the kth day.

The first probability is just the CDF of the geometric distribution with parameter 1/3 evaluated at k=3.

I'll leave you to show that this is $p=\dfrac {63}{81}$

Now using this $p$ we find the probability that we succeed in lighting the fire in under 4 tries on the 3rd day.

This is just $(1-p)^2 p = \dfrac{16640}{531441}$
 
Dec 2015
25
1
Worcester, UK
Thanks, but - I forgot to mention this in my post, my bad - I knew that using P(X>4) and letting this equal p and then doing \(\displaystyle (1-p)^{2}p\) will result in the answer, but I don't understand why just calculating P(X=3) doesn't work.
 

romsek

MHF Helper
Nov 2013
6,641
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California
because P(X=3) is the probability that on a given day it takes you 3 tries to light the stove, which isn't at all the same as what they are asking for.
 
Dec 2015
25
1
Worcester, UK
Hmm..

I'm struggling to see why spreading out the attempts over longer periods of time invalidates P(X=3).
 

romsek

MHF Helper
Nov 2013
6,641
2,993
California
Hmm..

I'm struggling to see why spreading out the attempts over longer periods of time invalidates P(X=3).
There is a small glitch in the language of the problem. I see where your confusion comes from.

The problem says Henry must light the fire once a day.

I interpret this as he must successfully light the fire. Not just attempt to light it. That means once a day he has to try to light the fire until successful.

You have interpreted this sentence as once a day he tries to light the fire, successful or not.

My solution is for the first case. Yours is for the second.
 
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Dec 2015
25
1
Worcester, UK
OHHHHHHHHH... Thanks so much! You're right, I misread "Henry has to light the fire once a day" as "Henry has to attempt to light the fire once a day" (which, in hindsight, is ridiculous: why would he need to wait 24 hours before making another attempt!)
 
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