When is possession established on a jump ball

Question
Game being played under NFHS rules, at the beginning of the game the jump ball is tipped by team A towards their offensive basket. The ball hits the floor in bounds but rolls out of bounds without any player touching the ball. The referee awarded the ball to team B (team A last touched the ball before going out of bounds). The referee awarded the possession arrow to team B because team A won the jump. Should the ball have been re-jumped? What is the correct ruling(s)? Would the rulings change if using NCAA rules?

Answer
Hi Keith,

The call was partially correct and partially incorrect. The ball was tipped by Player A before going out of bounds, and Player A was the last to touch it, so Team B should be awarded a designated throw in.  However, The Alternating Possession Arrow should have been awarded to Team A not Team B. The reason being, Team A tipped the ball, but never gained possession of the ball. Team B had the first possession by virtue of the out of bounds call.
There would be no need for a re-jump either, as the play was completely legal. The same rules would apply to NCAA rules as well, as far as I know,  since they use the alternating possession rule as well.  The NBA would be different as they do not use the alternating possession Arrow.  I hope that cleared it up for you.