1.

Do we need to mention that ownerships of opportunity and ervolver are independent?

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4.

a) and b) are sort-of similar. 
And for b), seems that the key part is to realize that 
more than 9 flips means branching back to step 1 for 3 times.

I wonder if we can ask a question related to the random variable $k$?
Such as... given step 3, show that  Pr(k=1) = 1/3.

c)
I think that should be
 3/ (1-p^3 - (1-p)^3)

(Also, this is quite closely related to a) and b), IMO)

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5.

typo: Let k be _an_ integer ...

Maybe {1,2,3, ... , p_n!}  (just to be a bit more obvious)


b) Counter Example:
n=2
p_2=3

set: {1,2,3,4,5,6}

p_{3}=5

But 5|5

So it's possible that p_{n+1}|k
Unless I'm misunderstood something?

c) Same problem if my understanding of b) is not mistaken...