The box still has 52 cards; relabeling does not change their quantity.
There are 13 hearts and four Jacks, but one of the Jacks is a heart, so it will be counted twice in the sum 13 + 4. Therefore the number of relabeled cards that have a 1 is 13 + 4 -1 = 16. The number that have a zero must then be 52 - 16 = 36.
The probability of drawing a 1 is therefore 16/52 = 4/13. This is the same as the probability of drawing either a heart or a Jack from a standard deck at random.
1. One name for the relabeled box is B(1, 4/13). In general, a box with k 1's and n-k 0's will have n tickets total. The probability of a 1 is k/n. The name of this box is B(1, k/n). That is a shorthand for "Binomial" (or "Binary" or "Bernoulli"), 1 trial, probability k/n of success.
2. This form of relabeling, where a "1" indicates an event of interest and a "0" indicates all tickets not of interest, is called a "characteristic function." It is useful for computing probabilities of complex events (sets of tickets), because it clearly reduces the computation to a counting problem.