Explaining mixed strategies for one-shot games In the classic introduction to non-cooperative game theory, the mixed strategy for a player is taught as a distribution over strategy space for the player. The distribution essentially gives us the probabilities (say, discrete strategy set) with which a player should play the strategies in a Nash equilibrium. 

However probabilities carry the notion of being frequencies and these essentially mean the long-run fraction of games in which the player should play the strategy. However the setting is a one-shot game and this is a contradiction.

How do we resolve the contradiction when explaining what a mixed strategy is?
Ariel Rubinstein tends to be insightful regarding these kinds of questions. 

He addresses the interpretation of mixed strategies in section 3 of this paper. 

A few possible interpretations aside from deliberate randomization:


Purification: A mixed strategy is a plan of action based on information not specified in the model.
A fictitious long run story.
Population averages, so imagine the player's being pulled from some population distribution where different types play different pure strategies. The  population distribution is the mixed strategy distribution.


An interesting quote regarding player $i$'s mixed strategy reflecting uncertainty among $-i$'s regarding what $i$ will do:


  Mixed strategy can alternatively be viewed as the belief held by all other 
  players concerning a player's actions. A mixed strategy equilibrium is then an 
  n-tuple of common knowledge expectations, which has the property that all the 
  actions to which a strictly positive probability is assigned are optimal, given the 
  beliefs. A player's behavior may be perceived by all the other players as 
  the outcome of a random device even though this is not the case. Adopting this 
  interpretation requires the reassessment of much of applied game theory. In 
  particular, it implies that an equilibrium does not lead to a prediction (statistical 
  or otherwise) of the players' behavior. Any player i's action which is a best 
  response given his expectation about the other players' behavior (the other 
  n - 1 strategies) is consistent as a prediction for i's action (this might include 
  actions which are outside the support of the mixed strategy). This renders 
  meaningless any comparative statics or welfare analysis of the mixed strategy 
  equilibrium and brings into question the enormous economic literature which 
  utilizes mixed strategy equilibrium.