I am going through an introductory course on game theory, and on a lecture slides it reads:
"A mixed strategy is a probability distribution on the set of pure actions, for example, Prisoner A confesses with probability 0.4 and betrays with probability 0.6"
I am taking a probability course at the same time, where probability distribution refers to a specific probability density function. If a random variable $X$ has a Gaussian distribution, then $X \sim \mathcal{N}(m, \sigma^2).$ It can have many other types of distribution, uniform, geometric, Poisson, Weibull, etc. ...
So what does it mean by a "A mixed strategy is a probability distribution on the set of pure actions"?
Does it mean that the mixed strategy is a random variable $X$ that follows a specific probability density function $f_X(x)$? If so, what probability density function? What are the event space for which the random variable is mapping from?
It's as simple as in the given example.
When it comes to choose the best strategy one first has to define the set of feasible strategies. In some cases we just have a finite set ${\cal S}=\{S_1,S_2,\ldots, S_n\}$ of possible strategies, and we have to choose the best of these; period. In more involved cases (e.g., if the game in question will be played many times) a strategy consists of a probability distribution on ${\cal S}$. This means that we have to choose $n$ probabilities $p_i\geq0$ summing to $1$ beforehand, and then will select the $S_k$ "at runtime" according to these probabilities. Such strategies are called mixed strategies. The problem now consists in finding the best mixed strategy. This is the probability distribution ${\bf p}=(p_1,p_2,\ldots,p_n)$ on ${\cal S}$ for which the expected gain is largest.