# Does the propensity interpretation of probability rely on the principle of indifference?

According to the late Popper, among others, probability is the propensity of a set of conditions to produce certain long run relative frequencies. Therefore if we say that a certain set of conditions have the probability of heads of 0.5, we thereby express that repeatedly conducting an experiment according to these conditions will yield a long run frequency of heads of 0.5. This does not mean that the propensity is the same as the long run frequency, but rather that the propensity is what causes the long run frequency to behave in that manner.

"In this interpretation, the generating conditions are considered as endowed with a propensity to produce the observed frequencies. As Popper put it: ‘But this means that we have to visualise the conditions as endowed with a tendency or disposition, or propensity, to produce sequences whose frequencies are equal to the probabilities; which is precisely what the propensity interpretation asserts.’" (Popper quoted by Gillies in "Philosophical Theories of Probability").

I wonder whether the propensity interpretation implicitly utilizes something like the principle of indifference. I will illustrate my thought process by the means of a concrete example:

I can characterize a coin toss according to conditions C, such as:

• the coin is tossed by a person with no intention to influence its outcome.
• the coin is symmetric.
• the coin is tossed on earth.
• etc.

Now assume now that I have data D of 1000 such experiments that were conducted under the above explicated conditions. My best estimate of the propensity is then presumable the relative frequency of heads in these 1000 experiments. So P(heads|condition C)= number of heads/1000 := h/n

So assume now that I am in a situation that can be characterized according to the above conditions C and I want to determine the single case probability for the situation. My "best guess" for the propensity then seems to be h/n, if I have not further relevant information about how the coin toss is conducted apart from the fact that it was conducted under conditions C.

But what is actually done when h/n is calculated? Every data point in the data set D follows the characterization C. But obviously the conditions for each toss beyond that were not equal. Otherwise the results would have been the same for all 1000 experiments. The conditions varied in some way that was not measured. However when I calculate h/n I weigh the outcome of each experiment equally. It seems to me however that this is something very similar to the principle of indifference. Weighing each experiment equally corresponds to a believe that each experiment was equally "entitled".

The same argument would hold e.g. if I was to determine the weight of an object. I might weigh a specific object 1000 times. My best guess would then be that the true weight of the object is the sum of all measurements divided by 1000. But this again utilizes the principle of indifference in a similar manner than in the above example. Because I have no reason whatsoever to prefer one measurement over the other I weigh them equally.

Would you agree that this is the principle of indifference in disguise or am I profundly confused about the matter?

I want to add another question that was inspired by the comments. In many examples, such as roulette or coin tosses I think the propensity of a situation only makes sense because the conditions C do not completely describe the situation: in the coin toss example, the exact angle and motion etc. are not specified. The set of generating conditions is in some sense incomplete. So it seems to me that propensity in such situations does not actually refer to a physical property. The fact that the same conditions can result in different outcomes is merely due to the fact that the conditions to not completely characterize the situation. Is this correct?