I have two dogs and they're very neurotic. Mingus isn't happy unless at least 2 of his 3 favorite chewtoys are in the backyard with him, and Duke isn't happy unless at least 3 of his 5 favorite chewtoys are in the frontyard with him.
The probability that any particular chewtoy is not in its respective dog's yard is 1-a. (Note that the probability is independent from chewtoy to chewtoy, but the probabilities themselves are all the same). For what values of a is Mingus' happiness more probable than Duke's happiness?
What I think:
Intuitively, I want to say that since the probabilities are all the same that it's always a safer bet on Mingus' happiness over Duke's, but I've learned that my intuition sucks at probability. So I tried calculating the probability that each dog will be happy. I'll use my Mingus calculation as an example:
Prob(Mingus happy) =
=1-Prob(Mingus not happy)
= 1-Prob(2 Mingus toys missing or 3 Mingus toys missing)
= 1-( a(1-a)^2 + (1-a)^3 ) = 2a-a^2 (if I did my algebra right)
I also tried:
=Prob(2 Mingus toys in yard or 3 Mingus toys in yard)
=a^2 * (1-a) + a^3 = a^2.
So I have two issues, and any clarification on these would be awesome:
(1) I got something different when I tried to calculate the same thing two different ways, and
(2) by either way of doing, I set Prob(Mingus happy) > Prob(Duke happy) and tried to solve for a range of values for a but got either something unhelpful or an inequality that agreed with my earlier intuition.