The equation for finding sample size with a given proportion is given by (see this link for details):

where p is the true proportion, Z is the z-value for a given confidence level, and e is the sampling error. So consider the information you're given. We know that since the coin is unbiased that half of the tosses should end up on one side or the other. Thus, p = 0.5, and p(1-p) = 0.25. Your error is the difference between the observed and the true proportion. The question states that we want at least 75% of heads to be within 0.1 of the true proportion. Thus, the error is just that difference. The last piece of information you require for your calculation is Z for an . I get that because the confidence level just is . For instance, standard confidence levels are to find the observed at least 95% of the time with an , which gives a Z = 1.96. In our case, we should get Z = 1.15.

I get a value of n just slightly above 33. Since these are discrete, we have to round up to whole numbers.