You have to hand it to great apes

Mothers cradle their infants with the left, rather than the right, arm. So do daddies. Babies too, seem happier when on their parents’ left. Why is that side preferred? Some candidate explanations come to mind.
Most mothers are right-handed; cradling to the left frees the more dextrous right limb to minister to the child. Against that, it would be easier to carry an infant with the stronger right arm.