You want to send data from your phone to your home server.
There are two possible communication methods: a fast but unreliable wifi network or a slow but more reliable bluetooth connection. Your phone attempts to send the data at the start of every second and it arrives at the end of that second if successfully transmitted over wifi, but arrives at the end of the following second if successfully transmitted over bluetooth. For each attempt to send data the probability of successful transmission is 0.05 via wifi and 0.5 via bluetooth. The phone can only attempt one communication method each second, either wifi or bluetooth.
There is a user control, however, to set the probability w that data should be sent by wifi. For example, if w = 1, the data is always sent by wifi, if w = 0, the data is always sent by bluetooth.
Note that the fastest possible arrival time is one second if the first attempt is to send the data by wifi and it is successful. Note also that simultaneous arrival of data by the two different methods does not cause any delays in the server.
You prefer to send the data by wifi, but do not want the expected arrival time of the data to be greater than 5 seconds. Find the maximum value of w that could be set.
Please enter your answer to three decimal places.