Assume that there is a sequence of individuals, each deciding whether to adopt or reject some behavior. Each individual observes the decisions of all those ahead of him. The ordering of individuals is exogenous and is known to all.
All individuals have the same cost of adopting, C, which for now we set to 1/2. The gain to adopting, V, is also the same for all individuals and is either zero or one, with equal prior probability 1/2. Individuals differ in their positions in the queue. Each individual privately observes a conditionally independent signal about value. Individual i's signal X_i is either H or L, and H is observed with probability p > 1/2 if the true value is one and with probability 1 - p if the true value is zero, L is observed with probability p>1/2 if the true value is zero and 1-p if the true value is one.
E.g. The occurrence of HL or LH involves a coin flip, so the total probability is 1/2(p)(1-p) + 1/2(p)(1-p) = p(1-p).
What is the probability of the occurrence of HH, and LL?