All elements that have more than 83 protons in their nucleus are radioactive, meaning that they will give off energy in the form of radiation and particles until they degrade into more stable forms of elements. The two you asked about, iodine, atomic number 53, and barium, atomic number 56, both fall under the cutoff line for natural radioactivity. One way you can tell if an element is radioactive or not is with the use of a Geiger Counter, which measures the nuber of nuclei being degraded per minute. Geiger Counters measure this activity with "clicks"; the more clicks you hear coming from the counter means a higher rate of radioactivity. I think you have your number wrong on the "barium-13" example question, because the mass number of barium, which is the number of protons plus the number of neutrons, is 137. Picking between the two of them, barium is higher in atomic nuclei number on the periodic chart, so I would pick barium in terms of instability, and iodine in terms of stablility.