A trial of the random experiment corresponds to taking a random walk from the root ( the top node in the tree, where no coins have been flipped ) to a leaf.
12.
To apply the method to a probabilistic proof, the randomly chosen object in the proof must be choosable by a random experiment that consists of a sequence of " small " random choices.
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The basic idea is to replace each random choice in a random experiment by a deterministic choice, so as to keep the conditional probability of failure, given the choices so far, below 1.
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The method of conditional probabilities replaces the random root-to-leaf walk in the random experiment by a deterministic root-to-leaf walk, where each step is chosen to inductively maintain the following invariant:
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For two elements x, y \ in \ mathcal X the random variables Z ( x ) and Z ( y ) will not be uncorrelated, because if x is too close to y the random experiments described by Z ( x ) and Z ( y ) will often show similar behaviour.
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