Where Will the Bikes Be?

A city bike-share scheme has $n$ docking stations. Months of trip logs show how a single bike tends to move during one day:

• P[i][j] is the probability that a bike which starts the day at station i ends that day at station j.
• Every row of P sums to $1$, because a bike always ends the day somewhere.
• Each bike moves independently of the others, and where it goes tomorrow depends only on where it is tonight (the Markov property).

Tonight the fleet is fleet[i] bikes at station i. To plan the redistribution truck, the operator wants the expected number of bikes at each station after days more days.

Task: write fleet_forecast(P, fleet, days) returning a list with the expected number of bikes at each station after exactly days days, each rounded to 4 decimal places.

• days is a whole number, 0 or more. After 0 days nothing has moved yet.
• P is square and its size matches fleet. Expected counts can be fractional, and that's fine.

Once your function works, try one P with a large days and two very different starting fleets, and compare the two answers.