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Long time reader first time poster...

This simple question has stumped me all day and I think I've finally cracked it! I'm hoping someone can confirm that, or tell me how wrong I am - either is fine :)

*One in 1000 cows have a rare genetic disease. The disease is not contagious, therefore cases are independent.*

Let X be the number of cows purchased by a farmer. How many cows are purchased by the farmer until the 1st cow with the disease, given:

P(X≤r)=0.05

P(X≤r)=0.90

Let X be the number of cows purchased by a farmer. How many cows are purchased by the farmer until the 1st cow with the disease, given:

P(X≤r)=0.05

P(X≤r)=0.90

This is what I've done:

p = 1/1000 = 0.001 (? was unsure if this was in fact my p value)

P(X>r)=(1-p)^r

P(X≤r)=0.05 (given)

P(X≤r) + P(X>r) = 1 for geometric distribution

Therefore:

0.05 + (1-p)^r=1

0.05 + (1-0.001)^r=1

0.999^r=0.95

ln(0.999^r)=ln(0.95)

r≈51

And same again for P(X≤r)=0.90

Can someone tell me if I'm heading in the right direction - or is there a better way?

Thanks