Originally Posted by

**cribby** Problem:

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(Mingus happy)=

=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.