Could someone explain the intuition here.
If A is the event that at least one couple has the characteristics, why is the event A listed as the intersections of all the Ai's not happening?
By extension I do not see what B represents here either. I understand C is the Probability that exactly one couple has the characteristics.
I would really appreciate your help.
This is the question I am referring to, it is from Statistics by DeGroot page 71
On
Event A is atleast one couple has the characteristics is same as from probability of all possibilities substrate when none of couple has characteristics. i.e P(A)=1-$\;{(1-p)}^n$. Event B is atleast 2 of the couple has characteristics hence from atleast one i.e from event A if we remove exactly one couple ( i.e event C ) has characteristics we get B. P(B)=P(A)-P(B). = 1-${(1-p)}^n\;-\;np{(1-p)}^{(n-1)}$.
Look again: $$A=(A_1^\complement\cap A_2^\complement\cap\cdots\cap A_n^\complement)^{\color{red}\complement}$$ so $A$ is (correctly) written as the opposite of all couples not having the property, which is just a more manageable rewording of at least one copuple has the property. Now with $C$ representing exactly one couple has the property, $B=A\cap C^\complement$ represents the case of at least one but not exactly one, i.e., at least two.