A message being sent over a communications network is assigned

by a router to one of three paths $($path $1,$ path $2,$ path $3).$ The nature

of the network is such that $50\%$ of all messages are routed to path $1,$

$30\%$ are routed to path $2,$ and $20\%$ are routed to path $3.$ If routed to

path $1,$ then the message has a $75\%$ chance of reaching its destination immediately. Otherwise, the message experiences a five$-$second

delay and returns to the router. If routed to path $2,$ then the message

has a $60\%$ chance of reaching its destination immediately. Otherwise,

the message experiences a ten$-$second delay and returns to the router.

If routed to path $3,$ then the message has a $40\%$ chance of reaching

its destination immediately. Otherwise, the message experiences a

twenty$-$second delay and returns to the router. Note that the router

cannot distinguish between new messages and messages that have

returned from an unsuccessful attempt. Let X denote the time until

the message reaches its destination. Determine E$($X$)$ and StDev$($X$).$