Looking at the extended forecast on weather.com, they show only a 60% chance of showers on Thursday, January 4.
Not to get your hopes up, but rainy weather forecasts for Southern California are notoriously unreliable. They'll tell us there's a "chance of rain" when it's already pouring outside; or they'll have us preparing for "the Mother of All Storms" which then turns out to be nothing more than a light drizzle. (Frequently, storms from the north lose power and/or turn inland before making it down this far.)
Today's forecast in the L.A. Times, for example, says that a "strong Alaskan storm system" will be "generating scattered heavy showers." But from where I sit (near LAX), the sun is shining and any clouds I see are far away to the northeast.
Enjoy your trip regardless!