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JWST transmission and emission spectra will provide invaluable glimpses of transiting exoplanet atmospheres, including possible biosignatures. This promising science from JWST, however, will require exquisite precision and understanding of systematic errors that can impact the time series of planets crossing in front of and behind their host stars. Here, we provide estimates of the random noise sources affecting JWST NIRCam time-series data on the integration-to-integration level. We find that 1/f noise can limit the precision of grism time series for 2 groups (230 ppm to 1000 ppm depending on the extraction method and extraction parameters), but will average down like the square root of N frames/reads. The current NIRCam grism time series mode is especially affected by 1/f noise because its GRISMR dispersion direction is parallel to the detector fast-read direction, but could be alleviated in the GRISMC direction. Care should be taken to include as many frames as possible per visit to reduce this 1/f noise source: thus, we recommend the smallest detector subarray sizes one can tolerate, 4 output channels and readout modes that minimize the number of skipped frames (RAPID or BRIGHT2). We also describe a covariance weighting scheme that can significantly lower the contributions from 1/f noise as compared to sum extraction. We evaluate the noise introduced by pre-amplifier offsets, random telegraph noise, and high dark current RC pixels and find that these are correctable below 10 ppm once background subtraction and pixel masking are performed. We explore systematic error sources in a companion paper.