The consequences on transcription of BUdR substitution of the normal thymidine in nuclear DNA are analyzed from a probabilistic point of view. The probability that at any stage subsequent to the administration of BUdR, a randomly chosen cell will be able to make good transcripts is calculated both under the assumption that incorporation of BUdR is complete and under the assumption that incorporation is partial. In the latter case both gradually increasing and decreasing incorporation rates are considered. The presented calculations may be of value for the investigation of any type of differentiating cells. Using the example of the sea urchin embryo it is shown how to calculate the probability that a particular group of cells, which in further development differentiate to a certain organ, will inherit as sufficient number of essential genes unaffected by BUdR in order to ensure a normal development. It is shown that the probability that the four macromeres (or micromeres) in the vegetal half of the 16-cell stage of the sea urchin embryo will contain at least one, two or four plus (or minus) strands of DNA uninfluenced by BUdR gradually decreases the earlier BUdR is administered. The theoretical calculations are in agreement with experimental observations on sea urchin embryos (described in part II).