(Introduction to Probability, Blitzstein and Nwang, p.80)
Alice, Bob, and 100 other people live in a small town. Let C be the set consisting of the 100 other people, let A be the set of people in C who are friends with Alice, and let B be the set of people in C who are friends with Bob. Suppose that for each person in C, Alice is friends with that person with probability 1/2, and likewise for Bob, with all of these friendship statuses independent.
(a) Let D $\subseteq$ C. Find P (A = D).
(b) Find P (A $\subseteq$ B).
(c) Find P (A $\cup$ B = C).
My general approach to those problems was conditioning on the set sizes. Part b) must be certainly false.
a)
$P(A=D) = \sum_{k=0}^{100} P(A=D \mid |A|=k) * P(|A|=k)$
$P(|A|=k) = \binom{100}{k} \left(\frac{1}{2}\right)^{100}$
\begin{align} P(A=D \mid |A| = k) &= \sum_{j} P(A=D \mid |A| = k, |D|=j) * P(|D|=j)\\ &= P(A=D \mid |A|=|D|=k) * P(|D|=k)\\ &= \binom{100}{k}^{-1} * \frac{1}{101} \end{align}
\begin{align} \sum_{k=0}^{100} \binom{100}{k}^{-1} * \frac{1}{101} * \binom{100}{k} * \left(\frac{1}{2}\right)^{100} = \left(\frac{1}{2}\right)^{100} \doteq 0 \end{align}
b) \begin{align} P(A \subseteq B) &= \sum_{k\leq j} P(A \subseteq B \mid |A| = k, |B|=j) * P(|A| = k, |B|=j)\\ &= \sum_{k\leq j} \frac{\binom{k}{j}}{\binom{100}{k}\binom{100}{j}} * \binom{100}{j} \left(\frac{1}{2}\right)^{j} * \binom{100}{k} \left(\frac{1}{2}\right)^{k}\\ &= \sum_{k\leq j} \binom{k}{j} \left(\frac{1}{2}\right)^{k+j} = 4 \end{align}
c)
\begin{align} P(A \cup B = C) &= \sum_{k=0}^{100} P(A \cup B = C \mid |A|=k, |B|=100-k) * P(|A|=k, |B|=100-k)\\ &= \sum_{k=0}^{100} \frac{1}{\binom{100}{k}} * \binom{100}{k} \left(\frac{1}{2}\right)^{100} * \binom{100}{100-k} \left(\frac{1}{2}\right)^{100}\\ &= \sum_{k=0}^{100} \binom{100}{k} \left(\frac{1}{2}\right)^{200} \doteq 0 \end{align}
Am I principally doing the right thing? Any help with this?