In an office there are four staff members. Their email system has the following function: if a staff member is on holiday and an email is sent to him/her, then the other three staff members will each receive the forwarded email, independently, with probability \(\frac{1}{2}\).

Suppose that one day all staff members are on holiday and a new email is sent to one of them. What is the probability that the system keeps forwarding the email among these four staff members indefinitely?

Please enter your answer as a decimal accurate to three decimal places.

Mathsbombe Competition 2018 is organised by the The Department of Mathematics at The University of Manchester.
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