The probability that a visit to a primary care physician’s (PCP) office results in neither lab
work nor referral to a specialist is 35%. Of those coming to a PCP’s office, 30% are referred to
specialists and 40% require lab work.
Calculate the probability that a visit to a PCP’s office results in both lab work and referral to
a specialist. (5)

You want the probability of visits that result in both referral and lab work if 30% of visits result in referral, 40% result in lab work, and 35% result in neither.

Neither

The case of "neither" can be written two ways:

P(R' · L') = (P(R) +P(L) -P(R · L))'

0.35 = 1 -(0.30 +0.40 -P(R·L)) = 0.30 +P(R·L)

Solving for P(R·L), we find ...

P(R·L) = 0.35 -0.30 = 0.05 = 5%

The probability that an office visit will result in both referral and lab work is 5%.