TL;DR - Sounds reasonable. Here’s why:
▫️In a world of people losing $5 is not a big deal, the likelihood of people finding $5 makes them happy is 63%, and people finding $5 doesn’t make them happy is 37% or about 1 in 3.
▫️People finding $5 makes them happy are 2x as likely to be people losing $5 is a big deal as people finding $5 doesn’t make them happy.
▫️To change our beliefs, the prevalence of people finding $5 makes them happy, whether or not they are people losing $5 is not a big deal, would need to be under 64%, and people finding $5 doesn’t make them happy would need to be over 36%, all else being equal.
Let’s say the prevalence or prior probabilities for people finding $5 makes them happy is 75.0000% (odds of 3.0000x or chances of 1 in 1.3333), and people finding $5 doesn’t make them happy is 25.0000% (0.3333x or 1 in 4.0000), whether or not they are people losing $5 is not a big deal.
Of the people finding $5 makes them happy, 40.0000% (0.6667x or 1 in 2.5000) are people losing $5 is not a big deal, let’s say, and 60.0000% (1.5000x or 1 in 1.6667) are people losing $5 is a big deal.
Of the people finding $5 doesn’t make them happy, 70.0000% (2.3333x or 1 in 1.4286) are people losing $5 is not a big deal, let’s say, and 30.0000% (0.4286x or 1 in 3.3333) are people losing $5 is a big deal.
Thus, people finding $5 makes them happy is 0.5714 times as likely to be people losing $5 is not a big deal as people finding $5 doesn’t make them happy.
Also, finding $5 makes them happy is 2.0000 times as likely to be people losing $5 is a big deal as people finding $5 doesn’t make them happy. We know this as the likelihood ratio, risk ratio, or Bayes Factor.
The Relative Risk Increase (Reduction) is -0.4286, and the Absolute Risk Increase (Reduction) is -30.0000% (0.2308x or 1 in 3.3333).
The prevalence of people losing $5 is not a big deal, or people losing $5 is a big deal, regardless of people finding $5 makes them happy or people finding $5 doesn’t make them happy, is 47.5000% (0.9048x or 1 in 2.1053), and 52.5000% (1.1053x or 1 in 1.9048), respectively.
Therefore, which is more likely? In a world of people losing $5 is not a big deal, the posterior probability of people finding $5 makes them happy is 63.1579% (1.7143x or 1 in 1.5833), and people finding $5 doesn’t make them happy is 36.8421% (0.5833x or 1 in 2.7143).
In a world of people losing $5 is a big deal, the posterior probability of people finding $5 makes them happy is 85.7143% (6.0000x or 1 in 1.1667), and people finding $5 doesn’t make them happy is 14.2857% (0.1667x or 1 in 7.0000).
That's an Attributable Risk of -22.5564% (0.1840x or 1 in 4.4333). The Accuracy Rate (that is, “true-positive” and “true-negative”) is 37.5000% (0.6000x or 1 in 2.6667), and the Inaccuracy Rate (that is, ”false-positive” and “false-negative”) is 62.5000% (1.6667x or 1 in 1.6000).
The probability of people finding $5 makes them happy, and people losing $5 is not a big deal is 30.0000% (0.4286x or 1 in 3.3333). The probability of people finding $5 makes them happy, and people losing $5 is a big deal is 45.0000% (0.8182x or 1 in 2.2222). The probability of people finding $5 doesn’t make them happy, and people losing $5 is not a big deal is 17.5000% (0.2121x or 1 in 5.7143). The probability of people finding $5 doesn’t make them happy, and people losing $5 is a big deal is 7.5000% (0.0811x or 1 in 13.3333).
Sensitivity analysis:
What would the prevalence or prior probabilities for people finding $5 makes them happy, and people finding $5 doesn’t make them happy, whether or not they are people losing $5 is not a big deal, need to be such that in a world of people finding $5 makes them happy given people losing $5 is not a big deal, that both posterior probabilities are equally likely? In other words, we’d be indifferent? The prevalence of people finding $5 makes them happy would need to be 63.6364% (1.7500x or 1 in 1.5714), and people finding $5 doesn’t make them happy would need to be 36.3636% (0.5714x or 1 in 2.7500), all else being equal.
Similarly, what would the prevalence or prior probabilities for people finding $5 makes them happy, and people finding $5 doesn’t make them happy, whether or not they are people losing $5 is a big deal, need to be such that in a world of people finding $5 makes them happy given people losing $5 is a big deal, that both posterior probabilities are equally likely? In other words, we’d be indifferent? The prevalence of people finding $5 makes them happy would need to be 33.3333% (0.5000x or 1 in 3.0000), and people finding $5 doesn’t make them happy would need to be 66.6667% (2.0000x or 1 in 1.5000), all else being equal.
What would the consequent probabilities for people losing $5 is not a big deal given people finding $5 makes them happy, and people losing $5 is a big deal given people finding $5 makes them happy, need to be such that in a world of people finding $5 makes them happy given people losing $5 is not a big deal, that both posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of people losing $5 is not a big deal given people finding $5 makes them happy would need to be 23.3333% (0.3043x or 1 in 4.2857), and people losing $5 is a big deal given people finding $5 makes them happy would need to be 76.6667% (3.2857x or 1 in 1.3043), all else being equal.
What would the consequent probabilities for people losing $5 is not a big deal given people finding $5 makes them happy, and people losing $5 is a big deal given people finding $5 makes them happy, need to be such that in a world of people finding $5 makes them happy given people losing $5 is a big deal, that both posterior probabilities are equally likely? In other words, we’d be indifferent? The probability of people losing $5 is not a big deal given people finding $5 makes them happy would need to be 90.0000% (9.0000x or 1 in 1.1111), and people losing $5 is a big deal given people finding $5 makes them happy would need to be 10.0000% (0.1111x or 1 in 10.0000), all else being equal.
Using the Wald test, the relationship or association between people finding $5 makes them happy/people finding $5 doesn’t make them happy, and people losing $5 is not a big deal is statistically significant (n=10,000). Odds Ratio (OR) = 0.2857, p < .001, 95% Confidence Interval (CI) [0.2592, 0.3149]. We can say the same between people finding $5 makes them happy/people finding $5 doesn’t make them happy, and people losing $5 is a big deal. OR = 3.5000, p < .001, 95% CI [3.1758, 3.8573].