Final answer:
The best method to ensure 'yes/no' data is kept private while remaining useful when flipping a coin is A: Heads for a true answer, tails for a random answer. This is known as randomized response and it effectively preserves individual privacy and allows for accurate statistical analysis over many trials.
Step-by-step explanation:
If a coin is flipped, the option that would ensure "yes/no" data is private while still remaining useful is A: Heads for true answer, tails for random answer. This method, known as randomized response, is a means to preserve privacy when answering potentially sensitive questions. It provides a buffer that obscures whether the given answer is genuine or not on any single trial, but still allows for accurate estimation of the true proportions when looking at many trials.
With probability, by specifying that 'heads' will be used as the true answer and 'tails' as a random response (which could also be 'yes' or 'no' with equal chance), the respondent is given plausible deniability for each individual response, protecting their privacy. Over many responses, however, a statistical analysis can filter out the noise introduced by the random answers and estimate the actual proportion of 'yes' in the population.