# Independence over Lotteries

This lecture explains the independence over lotteries axiom of expected utility theory.

Takeaway Points

1. Let p be a probability between 0 and 1 and X, Y, and Z be outcomes (or probability distributions over outcomes). A preference is independent if and individual weakly prefers X to Y if and only if he prefers X with probability p and Z with probability 1 – p to Y with probability p and Z with probability 1 – p.
2. Notice that Z occurring with probability 1 – p is the same in both of the lotteries. Consequently, it should not have any bearing on an individual’s preference between the those lotteries. This is what independence requires.
3. While independence over lotteries is straightforward with outcomes, it gets trickier with compound lotteries (i.e., lotteries over lotteries). The next lecture shows an example where preferences can go wrong.

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