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Abstract: Start with an elementary coin flip (heads or tails, with probabilities p and 1-p, say) and some permutations acting on a sample space. Mix these ingrediaents just so and you can make algebras of operators acting on Hilbert spaces. By focusing on the algebra of 2x2 complex matrices, we can give a gentle introduction to the ground-breaking work of Murray and von Neumann. We shall investigate the operator algebras called "factors": why they are fundamental objects, and what is their classification scheme ("factor types") developed by Murray and von Neumann. Fortunately, by means of the coin-flip/permuations recipe we can make factors of all types and use the associated probabilities (there will be countably many coins!) to classify them. The latter is a challenging problem which has been a theme in the subject from its inception to the present day.