Suppose the following three bets;

Win $100 with probability 1/3 and lose $50 with prob. 2/3
Win $100 with probability 3/4 and lose $300 with prob. 1/4
Win $100 with probability 19/20 and lose $1900 with prob. 1/20
Out these three bets, a risk-averse individual will

Question

Suppose the following three bets;

Win $100 with probability 1/3 and lose $50 with prob. 2/3
Win $100 with probability 3/4 and lose $300 with prob. 1/4
Win $100 with probability 19/20 and lose $1900 with prob. 1/20
Out these three bets, a risk-averse individual will

BlackTom AI · Accepted Answer

Here is the question as given, with what is available to analyze:

Question:
Suppose the following three bets;

Win $100 with probability 1/3 and lose $50 with prob. 2/3
Win $100 with probability 3/4 and lose $300 with prob. 1/4
Win $100 with probability 19/20 and lose $1900 with prob. 1/20
Out these three bets, a risk-averse individual will

Answer options provided:
not choose any of the three bets if they were not coerced to do so

What follows are observations and reasoning about the situation and the missing options:

- First, we can compute the expected value (EV) of each bet, since EV is a common starting point for evaluating bets when risk neutrality is assumed. For a bet with outcomes x and y occurring with probabilities p and (1-p), EV = p*x + (1-p)*y.
  • Bet 1: +$100 with p = 1/3, and -$50 with probability 2/3. EV = (1/3)(+100) + (2/3)(-50) = 33.33... - 33.33... = $0. So this bet has zero expected value.
  • Bet 2: +$100 with p = 3/4, and -$300 with probability 1/4. EV = (3/4)(+100) + (1/4)(-300) = 75 - 75 = $0. Again, zero expected value.
  • Bet 3: +$100 with p = 19/20, and -$1900 with probability 1/20. EV = (19/20)(+100) + (1/20)(-1900) = 95 - 95 = $0. Yet again, zero expected value.
- So, all three bets have an EV of zero. From a purely EV perspective (risk neutrality), a person would be indifferent among the bets, since none of them has a positive expected payoff.
- Now, considering risk-averse preferences, the picture changes because risk-averse individuals care about the variability (risk) of outcomes in addition to the expected value. A concave utility function means that larger losses loom disproportionately in terms of utility, and larger gains offer diminishing marginal utility. Therefore, even with EV = 0, a risk-averse person is likely to reject bets that entail significant downside risk relative to the upside, especially the third bet which carries a very large potential loss (-$1900) for a modest probability.
- Let us reflect on the concept of risk premium: a risk-averse individual would typically demand a positive risk premium (i.e., require a positive expected net utility beyond zero EV) to compensate for the uncertainty and the possibility of large losses. Given the three bets, the downside in each bet grows dramatically in the third bet, making it particularly unattractive for someone who dislikes risk, despite the zero EV.
- The partial statement provided as a proposed answer—“not choose any of the three bets if they were not coerced to do so”—aligns with the intuition that a risk-averse person would refrain from taking bets that do not offer any expected gain and have substantial risk, especially when voluntary participation is possible. This reflects the standard view that risk-averse preferences favor avoiding bets with large downside risk when no positive EV is present.
- Regarding the missing answer options, since the list of answer choices is empty in the data you provided, there is nothing to evaluate against the concept above within the given options. In a typical multiple-choice setting, one would compare each alternative against the risk-averse reasoning just discussed, focusing on how each option treats expected value, risk, and utility.
- To summarize the embedded logic in practical terms: zero EV bets are not attractive to a risk-averse individual if there is any substantial downside, especially when participation is optional. Hence, choosing none of the bets would be a sensible stance for a risk-averse person in the absence of coercion or external incentives.
- If additional options were available, one would examine them for whether they imply taking any bet with zero EV or minimizing downside exposure, and whether they account for the scale of risk in each bet. The best choice would typically reflect both nonpositive EV and a preference to avoid large potential losses.
- In the current scenario, with no competing options to compare, the line of reasoning centers on the zero-EV nature of all bets and the risk aversion principle that large potential losses deter participation when there is no assured positive gain.

Suppose the following three bets; Win $100 with probability 1/3 and lose $50 with prob. 2/3 Win $100 with probability 3/4 and lose $300 with prob. 1/4 Win $100 with probability 19/20 and lose $1900 with prob. 1/20 Out these three bets, a risk-averse individual will Single choice

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