Good morning. The version of this you keep picturing has a probability attached, and it isn't one.
Decisions.
Expected Value
Think of the decision you are closest to committing to: could you state the four or five ways it could actually end, with a number against each and a payoff written in the same unit? Most people cannot — the whole thing has been running on one vivid outcome, and the branches that would compete with it have never been put anywhere you could argue with them.
Howard Raiffa, professor of managerial economics at Harvard, published Decision Analysis: Introductory Lectures on Choices under Uncertainty in 1968 and gave managers a way to hold an uncertain decision still long enough to look at it. Lay the decision out as a tree: the options in front of you, the outcomes each option could produce, a probability against every outcome and a value against every outcome. Then compare the options by expected value — the sum of each outcome multiplied by its odds. The arithmetic is trivial; the discipline is not, because the tree forces two things you would rather avoid. It forces enumeration, including the branches your preferred story leaves out. And it forces a number where you would sooner write an adjective. More than two centuries before that, in his 1738 paper on the measurement of risk, Daniel Bernoulli had already added the correction that still matters most: what a payoff is worth depends on what you already have, which is why expected value alone never settles a bet you could not survive losing.
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