Derive the probability that Tom W is a computer scientist.

You need:

$$P(B)=P(B|A)\cdot P(A) + P(B|A^C)\cdot P(A^C)$$

and:

$$P(B|A) = 4 \cdot P(B|A^C)$$

So then:

$$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}=$$

$$\frac{P(B|A)\cdot P(A)}{P(B|A)\cdot P(A) + P(B|A^C)\cdot P(A^C)}=$$

$$\frac{4 \cdot P(B|A^C) \cdot 0.03}{4 \cdot P(B|A^C) \cdot 0.03 + P(B|A^C)\cdot 0.97}=$$

$$\frac{0.12 \cdot P(B|A^C)}{P(B|A^C)\cdot (0.12+ 0.97)}=$$

$$\frac{0.12 }{1.09}\approx 0.11$$