A better way of thinking about this problem instead of adding N observations in a batch is adding them one at a time. The Beta(a,b) distribution does this - Uni

A better way of thinking about this problem instead of adding N observations in a batch is adding them one at a time. The Beta(a,b) distribution does this - Uniform is Beta(1,1). IIRC adding observations you're just ticking up a and b one at a time depending on whether the outcome was a head or a tails. This applies to exchangeable observations (if you flipped some other coin previously it doesn't work so simply in this model).