TL;DR — In a world of they did have sirens and car alarms to mock, the probability of birds would make completely different noises is 55.56% (1.25x or 100 for every 180), and birds wouldn’t make completely different noises is 44.44% (0.80x or 100 for every 225).

r/TheyDidTheMath

Let’s say the prevalence or prior probabilities for birds would make completely different noises is 66.67% (odds of 2.00x or chances of 100 for every 150), and birds wouldn’t make completely different noises is 33.33% (0.50x or 100 for every 300), whether or not they did have sirens and car alarms to mock.

In a world of birds would make completely different noises, 25.00% (0.33x or 100 for every 400) is they did have sirens and car alarms to mock, let’s say, and 75.00% (3.00x or 100 for every 133) is they didn’t have sirens and car alarms to mock.

In a world of birds wouldn’t make completely different noises, 40.00% (0.67x or 100 for every 250) is they did have sirens and car alarms to mock, let’s say, and 60.00% (1.50x or 100 for every 167) is they didn’t have sirens and car alarms to mock.

Thus, birds would make completely different noises is 0.63x as likely to be they did have sirens and car alarms to mock as birds wouldn’t make completely different noises.

Also, birds would make completely different noises is 1.25x as likely to be they didn’t have sirens and car alarms to mock as birds wouldn’t make completely different noises. We know this as the Likelihood Ratio, Risk Ratio, or Bayes Factor.

The Relative Risk Increase (Reduction) is -37.50%, and the Absolute Risk Increase (Reduction) is -15.00% (0.13x or 100 for every 667).

The prevalence of they did have sirens and car alarms to mock, or they didn’t have sirens and car alarms to mock, regardless of birds would make completely different noises or birds wouldn’t make completely different noises, is 30.00% (0.43x or 100 for every 333), and 70.00% (2.33x or 100 for every 143), respectively.

Therefore, which is more likely? In a world of they did have sirens and car alarms to mock, the posterior probability of birds would make completely different noises is 55.56% (1.25x or 100 for every 180), and birds wouldn’t make completely different noises is 44.44% (0.80x or 100 for every 225).

In a world of they didn’t have sirens and car alarms to mock, the posterior probability of birds would make completely different noises is 71.43% (2.50x or 100 for every 140), and birds wouldn’t make completely different noises is 28.57% (0.40x or 100 for every 350).

That's an Attributable Risk or Risk Difference of -15.87% (0.14x or 100 for every 630). The Accuracy Rate (that is, 'true-positive' and 'true-negative') is 36.67% (0.58x or 100 for every 273), and the Inaccuracy Rate (that is, 'false-positive' and 'false-negative') is 63.33% (1.73x or 100 for every 158).

The probability of birds would make completely different noises, and they did have sirens and car alarms to mock is 16.67% (0.20x or 100 for every 600). The probability of birds would make completely different noises, and they didn’t have sirens and car alarms to mock is 50.00% (1.00x or 100 for every 200). The probability of birds wouldn’t make completely different noises, and they did have sirens and car alarms to mock is 13.33% (0.15x or 100 for every 750). The probability of birds wouldn’t make completely different noises, and they didn’t have sirens and car alarms to mock is 20.00% (0.25x or 100 for every 500).

Sensitivity analysis:

What would the prevalence or prior probabilities for birds would make completely different noises, and birds wouldn’t make completely different noises, whether or not they did have sirens and car alarms to mock, need to be such that in a world where birds would make completely different noises given they did have sirens and car alarms to mock, and birds wouldn’t make completely different noises given they did have sirens and car alarms to mock, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The prevalence of birds that would make completely different noises would need to be 61.54% (1.60x or 100 for every 163), and birds that wouldn’t make completely different noises would need to be 38.46% (0.63x or 100 for every 260), all else being equal.

Similarly, what would the prevalence or prior probabilities for birds would make completely different noises, and birds wouldn’t make completely different noises, whether or not they didn’t have sirens and car alarms to mock, need to be such that in a world where birds would make completely different noises given they didn’t have sirens and car alarms to mock, and birds wouldn’t make completely different noises given they didn’t have sirens and car alarms to mock, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The prevalence of birds would make completely different noises would need to be 44.44% (0.80x or 100 for every 225), and birds that wouldn’t make completely different noises would need to be 55.56% (1.25x or 100 for every 180), all else being equal.

What would the consequent probabilities for they did have sirens and car alarms to mock given birds would make completely different noises, and they didn’t have sirens and car alarms to mock given birds would make completely different noises, need to be such that in a world where birds would make completely different noises given they did have sirens and car alarms to mock, and birds wouldn’t make completely different noises given they did have sirens and car alarms to mock, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of they did have sirens and car alarms to mock given birds would make completely different noises would need to be 20.00% (0.25x or 100 for every 500), and they didn’t have sirens and car alarms to mock given birds would make completely different noises would need to be 80.00% (4.00x or 100 for every 125), all else being equal.

Similarly, what would the consequent probabilities for they did have sirens and car alarms to mock given birds wouldn’t make completely different noises, and they didn’t have sirens and car alarms to mock given birds wouldn’t make completely different noises, need to be such that in a world where birds would make completely different noises given they did have sirens and car alarms to mock, and birds wouldn’t make completely different noises given they did have sirens and car alarms to mock, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of they did have sirens and car alarms to mock given birds wouldn’t make completely different noises would need to be 50.00% (1.00x or 100 for every 200), and they didn’t have sirens and car alarms to mock given birds wouldn’t make completely different noises would need to be 50.00% (1.00x or 100 for every 200), all else being equal.

What would the consequent probabilities for they did have sirens and car alarms to mock given birds would make completely different noises, and they didn’t have sirens and car alarms to mock given birds would make completely different noises, need to be such that in a world where birds would make completely different noises given they didn’t have sirens and car alarms to mock, and birds wouldn’t make completely different noises given they didn’t have sirens and car alarms to mock, that both these posterior probabilities are equally likely? In other words, we’d be indifferent? The likelihood of they did have sirens and car alarms to mock given birds would make completely different noises would need to be 70.00% (2.33x or 100 for every 143), and they didn’t have sirens and car alarms to mock given birds would make completely different noises would need to be 30.00% (0.43x or 100 for every 333), all else being equal.

Using the Wald test, the relationship or association between birds would make completely different noises or birds wouldn’t make completely different noises, and they did have sirens and car alarms to mock is statistically significant (n=10,000). Odds Ratio (OR) = 0.50, p < .001, 95% Confidence Interval (CI) [0.46, 0.55]. We can say the same between birds would make completely different noises or birds wouldn’t make completely different noises, and they didn’t have sirens and car alarms to mock. OR = 2.00, p < .001, 95% CI [1.83, 2.19].

Bayes’ Rule: P(H|E) = P(H) x P(E|H) / P(H) x P(E|H) + P(H’) x P(E|H’)

H birds would make completely different noises

H’ birds wouldn’t make completely different noises

E they did have sirens and car alarms to mock

E’ they didn’t have sirens and car alarms to mock

Contingency Table (n=10,000)