We have an additional piece of information that we can use: a white ball was selected from bag N.

Before we take the ball out, there is Sengnee's 2/7 chance of there being 5 red balls and 3 white balls (case one), and a 1-2/7=5/7 chance of there being 4 red balls and 4 white balls (case 2).

So taking a white ball out of case one has probability

and the probability of taking a white ball out of case 2 is

So knowing we took a white ball out, the probability that a red ball was transferred is the probability of taking a white ball out of case one divided by the total probability of taking out a white ball ie.