I agree when you say the odds of each flip are not affected by the other flips. But, I have observed a 75% likely aberration of flipping to above normal snowfall the year after well below normal snowfall. Two of those 4 above normal winters had a neutral Enso condition. One was a weak La Nina at - .9c. One was a weak El Nino at +.9c.
This past winter gave well below normal snowfall with a neutral to slightly positive Enso. at 0.0 to + 0.5 Enso based predictions would have been anomalously above actuality.
Next winter enso? We don't know, latest CPC probabilities issued March 19: 37% neutral, 31% La Nina, 32% El Nino