A student must choose one of the subjects, art, geology, or psychology

Let us say the probability that Art is chosen is x.
Then probability that Psychology is chosen is also x (since they are equally likely to be chosen)
The probability that Geology is chosen is 2x since it is twice as likely to be chosen.

Since, she must choose one of them, Probability (Art or Psychology or Geology) = 1 i.e. one subject of the three will definitely get chosen.
Also, she has to choose only one of them, not two or three. Hence Probability (Art or Psychology or Geology) is simply equal to P(Art) + P(Psychology) + P(Geology) (This is the most important step. Since two or more subjects cannot be chosen, there is no overlap in their probabilities. So the probability of choosing any one of them is simply equal to the sum of their individual probabilities. For more clarification, check out the theory of probability.)

1 = x + x + 2x
This gives us x = 1/4 = P(Art) = P(Psychology)
P(Geology) = 2x = 1/2

If all three subjects were equally likely to be chosen, what would be the probability of choosing Art?