Let's say I have 10 slots of which one of them is a special slot which we don't know. Each slot takes either 1 or 0 randomly and also each slot has 2 neighbor slots (neighbor slots are part of 10 slots itself).
Each slot is equally likely to have either 0 or 1. Special slot is chosen independently of contents.
A typical slot assignment is as follows. Here "7" is considered special slot.
There are $2^{10}$ data combinations possible.
Problem:
I would like to compute probability of selecting a slot that has opposite data to the neighboring slots and also that the chosen slot is the special slot.
By opposite data I mean, if the selected slot's data is 0, then the neighbor slots need to have 1 and 1. In other case if the selected slot's data is 1, then the neighbor slots need to have 0 and 0.
Based on my intuition this is what I thought of.
My Answer:
$E_1$=probability that a slot has opposite data to neighbor slots.
$E_2$=probability of choosing a worst slot.
$P(E_1) = 1/8 + 1/8 = 1/4$ (011 or 100 of a selected slot and its neighbor slots)
$P(E_2) = 1/10$ (choosing worst slot)
$P(E_1) P(E_2) = 1/40$
Is my approach correct?