Let us look at the tree again with the probabilities of the outcomes for which he tested positive appended.

					Outome	Probability
				B	Takes steroids & tests positive	0.095
					Takes steroids & tests negative
				B	Steroid-free & tests positive	0.135
					Steroid-free & tests negative
					Tests Positive	0.230

Since 23% of all rugby players test positive and, among that number, only 9.5% take steroids, the probability that a player who tested positive is taking steroids is

P(steroids|positive)

=

0.095 0.095 + 0.135

=

0.095 0.23

=

0.4130, or 41.3%