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What should you choose, a guaranteed $1\, 000\, 000$ dollars or flip a coin for a chance to win the following?

Heads: $0$, tails: $2 \,000 \,000$ dollars

Heads: $0$, tails: $5\, 000\, 000$ dollars

Heads: $0$, tails: $10 \,000 \,000$ dollars

Assume you only get ONE coin flip.

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    Hmmm... How much money is one supposed to have? ;P – 2017-01-19
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    "What would you do" is not a math question. "What should you do" is only a math question if a clear decision criterion is provided. Your post is off topic for this site. – 2017-01-19
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    Im not sure what the statistics that go into this are, but personally i would always go with the guaranteed. If we played the game with 10 vs 50 though, i might take the risk – 2017-01-19
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    @user2357112 no fun, but true – 2017-01-19
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    @user2357112 It is now – 2017-01-19
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    I'd choose a guaranteed million dollars in any case since 1 million would be life changing for me. If I had 50 million in the bank, I'd probably gamble on all 3, even though there's no real point to gambling on the first. If I had 10 million I'd probably gamble on the last one and not on the first two. Of course my estimates of what my risk preferences would be in these highly hypothetical scenarios could be off. – 2017-01-19
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    @TripleA Adding that you only get one flip makes it a more well-defined question but is not a decision criterion. There's still no mathematical answer to this question. – 2017-01-19
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    This is a game theory question, and belongs in an economics site. – 2017-01-19
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    How one would balance rational game theory against risk aversion is not a topic for this site. It's more of a psychological or economical subject, I'd say. – 2017-01-19
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    @Arthur I refuse to believe that, there is still an element of chance and probability, making it an equally viable question for this site. – 2017-01-20
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    Writing the question off as not suitable for this site, only makes me more curious about the answer, considering it is a tough one to figure out. – 2017-01-20
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    There is an element of chance and probability, yes, but the _question you're asking_ has nothing to do with that, and all to do with the irrationalities of risk aversion. What you should do depends greatly on your life situation, and without knowing more about that, there is no way to answer. I would probably keep the 1 million dollars. If I owed 5 million to the mafia, and they wanted their money back yesterday, I would definitely take the bet no matter what. If I had a grandmother dependent on expensive medical treatment, it would depend on exactly how expensive that medical treatment was. – 2017-01-20
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    @DougM I gave TripleA the math in my answer – 2017-01-20
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    @DougM Considering in maths, you usually disregard scenarios, I would assume, you are trying to achieve the highest sum of money from this event. – 2017-01-20
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    To see that a satisfactory answer to this question requires additional unstated assumptions that are not intrinsically mathematical, change the reward to $N, 2N, 5N, 10N$ instead of $N = 10^6$ dollars. Then, depending on $N$, the utility varies; specifically, the relationship between what a rational actor considers a worthwhile risk and the distribution of the payout will change based on an unstated relationship between $N$ and the amount of money $M$ that a person considers "large." But the question of what $M$ should be is a subjective one. – 2017-01-20
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    @TripleA If you would have given a utility function an appropiate answer could be given. – 2017-01-20
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    @callculus What exactly is that? – 2017-01-20
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    @jeff For instance a person has the utility function $U(x)=\sqrt{x}$, then the expected utility of the first choice would be $E(x)=0.5\cdot \sqrt 0+0.5\cdot \sqrt{5,000,000}=1118.03$. In contrast the (expected) utility of the garanteed alternative is $\sqrt{1,000,000}=1,000$ By using this concept the values are closer than in jeffs answer. This kind of calculation could lead to another choice as well. – 2017-01-20
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    @TripleA I have addressed my answer wrong. Please read the comment above. – 2017-01-20

2 Answers 2

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To start, I would calculate the expected value of all scenarios.

Scenario 1: Get a guaranteed $1 million.

The expected value would be: $1 \cdot 1\,000\,000$ dollars, or just a million dollars.

Scenario 2: Tails = $0$, Heads = $2\,000\,000$

The expected value here would be: $\frac 12 \cdot 0 + \frac 12 \cdot 2\,000\,000$, or still one million dollars, since a fair coin has the probability of landing on heads as $\frac 12$ and the probability of landing on tails as $\frac 12$.

Scenario 3: Tails = $0$, Heads = $5\,000\,000$

The expected value is $\frac 12 \cdot 0 + \frac 12 \cdot 5\,000\,000$, or $2\,500\,000$ dollars.

Scenario 4: Tails = $0$, Heads = $10\,000\,000$

The expected value is $\frac 12 \cdot 0 + \frac 12 \cdot 10\,000\,000$, or 5 million dollars.

But since I would much rather be guaranteed money rather than use probability, I would prefer the first scenario (the guaranteed $1 million), hence the answer to your question.

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This isn't even a viable choice to make.

Trivially the middle options are worse than the last.

And then it's a question of how diminishing money is for you. If 1kk is 1kk more than you have now, it's a trivial choice. If you have say, a house, a car, all mortgages cleared and have a nice life, then a flip for 3.5kk more ev is obviously good.