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I realize this question may not make any sense. I do understand the concept of expected outcome and how it can be used to guide decision making when you know the probabilities, costs, and benefits of certain outcomes.

But what if you don't know the probability of something happening, but rather you know that doing something changes the odds of something happening.

Let's say there is an outcome that is either true or false. There are many unknown variables that effect the outcome. The average probability of this event is also unknown.

If the outcome is true I win $3,000, if not I win nothing.

However, I have information on two actions I can take to increase the odds of a true outcome.

  • Action A: increases the odds of a true outcome by 2%, but costs $200

  • Action B: increases the odds of a true outcome by 5%, but costs $500

How do I calculate the option that if repeated over and over maximizes my profits in the long run (Prize money - costs for actions)

There are other variables that affect the outcome, but I do not know them in this scenario.

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Let $p$ be the probability of the outcome occurring. Then, under the baseline scenario when you do not take any action your expected profits, denoted by $E_0$, is $3000p$.

If you take action A then the expected profits denoted by $E_A$, $3000*1.02*p - 200$.

If you take action B then the expected profits denoted by, $E_B$, is $3000*1.05*p - 500$.

Thus, take Action A if $E_A > E_0$ and $E_A > E_B$,take Action B if $E_B > E_0$ and $E_B > E_A$ and do not take either action if $E_A < E_0$ and $E_B < E_0$.

Working through the inequalities implied by the above paragraphs you will get some bounds on $p$ which will determine what you should be doing which may be helpful in making a decision.

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  • $\begingroup$ Hint: $p\leq{1}$ is all you need. $\endgroup$ – GeoMatt22 Dec 20 '16 at 19:20
  • $\begingroup$ @GeoMatt22 I know I am being dense here, but could you elaborate a little more? $\endgroup$ – Michael Dec 20 '16 at 19:58
  • $\begingroup$ From the formulas of @Kant we have $\Delta{E_A}=E_A-E_0=30p-200$ and $\Delta{E_B}=E_B-E_0=150p-500$. Since $p<1$, we have $\Delta{E_A}\leq{-170}$ and $\Delta{E_B}\leq{-350}$. So taking options $A$ and/or $B$, you should expect to lose money. $\endgroup$ – GeoMatt22 Dec 20 '16 at 20:18

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