The Love Island villa has a very complex lighting system. For the swimming pool light to be on, a particular light in the bedroom and a particular light in the hallway must be on. There are identical switches in each of the bedroom and kitchen. As they are identical, it's not possible to determine which are on/off nor what they control.
The villa is cleaned every Monday morning, and all the switches in the bedroom are set to be on or off. All the switches are on with probability . In the kitchen, each individual switch is set to be on or off with probability , independently of all other switches. A new islander enters the villa after it's cleaned and selects one switch uniformly at random from both the kitchen and bedroom and flicks them in the opposite direction. Given that the light is on afterwards, find the probability that the new islander switched the swimming pool light in the bedroom. Evaluate this probability when .