Before a cricket match can begin the tradition is that the umpire tosses a coin and one of the captains calls, heads or tails, whilst the coin is in the air. If the captain gets it right s/he chooses which side opens the batting. There never seems to be an issue as to which captain actually makes this call (otherwise we would have to toss a coin and make a call to decide who makes the call, and in turn toss a coin and make a call to decide who makes that call and so on) since it seems clear that this procedure is fair. In other words both captains are giving equal probability to the coin landing heads as to it landing tails no matter which of them calls it. The obvious explanation for this is that both captains are, subconsciously perhaps, appealing to the symmetry of the situation.

At the same time they are, it seems, also tacitly making the assumption that all the other information they possess about the situation, for example the weather, the gender of the referee, even past successes at coin calling, is irrelevant, at least if it doesn't involve some specific knowledge about this particular coin or the umpires's ability to influence the outcome. Of course if we knew that on the last 8 occasions on which this particular umpire had tossed up this same coin the result had been heads we might well consider that that was relevant.

Forming beliefs, or subjective probabilities, in this way by considering symmetry, irrelevance, relevance, can be thought of as logical or rational inference. This is something different from statistical inference. The perceived fairness of the coin toss is clearly not based on the captains' knowledge of a long run of past tosses by the umpire which have favoured heads close to half the time. Indeed it is conceivable that this long run frequency might not give an average of close to half heads, maybe this coin is, contrary to appearances, biased.