Jurimetrics

In 2018, California's legislature passed Senate Bill 826, which requires all publicly held corporations based in the state to have a minimum number of women on their board of directors.[33][34] Boards with five or fewer members must have at least two women, while boards with six or more members must have at least three women.

Using the binomial distribution, we may compute what the probability is of violating the rule laid out in Senate Bill 826 by the number of board members. The probability mass function for the binomial distribution is:

$${\displaystyle \mathbb {P} (X=k)={n \choose {k}}p^{k}(1-p)^{n-k}}$$

where ${\displaystyle p}$ is the probability of getting ${\displaystyle k}$ successes in ${\displaystyle n}$ trials, and ${\textstyle {n \choose {k}}}$ is the binomial coefficient. For this computation, ${\displaystyle p}$ is the probability that a person qualified for board service is female, ${\displaystyle k}$ is the number of female board members, and ${\displaystyle n}$ is the number of board seats. We will assume that ${\displaystyle p=0.5}$. Depending on the number of board members, we are trying compute the cumulative distribution function:

$${\displaystyle {\begin{cases}\mathbb {P} (X\leq 1)=(1-p)^{n}+np(1-p)^{n-1},\quad &n\leq 5\\\mathbb {P} (X\leq 2)=\mathbb {P} (X\leq 1)+{n(n-1) \over {2}}p^{2}(1-p)^{n-2},\quad &n>5\end{cases}}}$$

With these formulas, we are able to compute the probability of violating Senate Bill 826 by chance:

As Ilya Somin points out,[33] a significant percentage of firms - without any history of sex discrimination - could be in violation of the law.

In more male-dominated industries, such as technology, there could be an even greater imbalance. Suppose that instead of parity in general, the probability that a person who is qualified for board service is female is 40%; this is likely to be a high estimate, given the predominance of males in the technology industry. Then the probability of violating Senate Bill 826 by chance may be recomputed as:

Bayes' theorem states that, for events ${\displaystyle A}$ and ${\displaystyle B}$, the conditional probability of ${\displaystyle A}$ occurring, given that ${\displaystyle B}$ has occurred, is:

$${\displaystyle \mathbb {P} (A|B)={\mathbb {P} (B|A)\mathbb {P} (A) \over {\mathbb {P} (B)}}}$$

Using the law of total probability, we may expand the denominator as:

$${\displaystyle \mathbb {P} (B)=\mathbb {P} (B|A)\mathbb {P} (A)+\mathbb {P} (B|\sim A)[1-\mathbb {P} (A)]}$$

Then Bayes' theorem may be rewritten as:

$${\displaystyle {\begin{aligned}\mathbb {P} (A|B)&={\mathbb {P} (B|A)\mathbb {P} (A) \over {\mathbb {P} (B|A)\mathbb {P} (A)+\mathbb {P} (B|\sim A)[1-\mathbb {P} (A)]}}\\&={1 \over {1+{1-\mathbb {P} (A) \over {\mathbb {P} (A)}}{\mathbb {P} (B|\sim A) \over {\mathbb {P} (B|A)}}}}\end{aligned}}}$$

This may be simplified further by defining the prior odds of event ${\displaystyle A}$ occurring ${\displaystyle \eta }$ and the likelihood ratio ${\displaystyle {\mathcal {L}}}$ as:

$${\displaystyle \eta ={\mathbb {P} (A) \over {1-\mathbb {P} (A)}},\quad {\mathcal {L}}={\mathbb {P} (B|A) \over {\mathbb {P} (B|\sim A)}}}$$

Then the compact form of Bayes' theorem is:

$${\displaystyle \mathbb {P} (A|B)={1 \over {1+(\eta {\mathcal {L}})^{-1}}}}$$

Different values of the posterior probability, based on the prior odds and likelihood ratio, are computed in the following table:

${\displaystyle \mathbb {P} (A|B)}$ with Prior Odds and Likelihood Ratio

	Likelihood Ratio
Prior Odds	1	2	3	4	5	10	15	20	25	50
0.01	0.01	0.02	0.03	0.04	0.05	0.09	0.13	0.17	0.20	0.33
0.02	0.02	0.04	0.06	0.07	0.09	0.17	0.23	0.29	0.33	0.50
0.03	0.03	0.06	0.08	0.11	0.13	0.23	0.31	0.38	0.43	0.60
0.04	0.04	0.07	0.11	0.14	0.17	0.29	0.38	0.44	0.50	0.67
0.05	0.05	0.09	0.13	0.17	0.20	0.33	0.43	0.50	0.56	0.71
0.10	0.09	0.17	0.23	0.29	0.33	0.50	0.60	0.67	0.71	0.83
0.15	0.13	0.23	0.31	0.38	0.43	0.60	0.69	0.75	0.79	0.88
0.20	0.17	0.29	0.38	0.44	0.50	0.67	0.75	0.80	0.83	0.91
0.25	0.20	0.33	0.43	0.50	0.56	0.71	0.79	0.83	0.86	0.93
0.30	0.23	0.38	0.47	0.55	0.60	0.75	0.82	0.86	0.88	0.94

If we take ${\displaystyle A}$ to be some criminal behavior and ${\displaystyle B}$ a criminal complaint or accusation, Bayes' theorem allows us to determine the conditional probability of a crime being committed. More sophisticated analyses of evidence can be undertaken with the use of Bayesian networks.

In recent years, there has been a growing interest in the use of screening tests to identify drug users on welfare, potential mass shooters,[35] and terrorists.[36] The efficacy of screening tests can be analyzed using Bayes' theorem.

Suppose that there is some binary screening procedure for an action ${\displaystyle V}$ that identifies a person as testing positive ${\displaystyle +}$ or negative ${\displaystyle -}$ for the action. Bayes' theorem tells us that the conditional probability of taking action ${\displaystyle V}$, given a positive test result, is:

$${\displaystyle \mathbb {P} (V|+)={\mathbb {P} (+|V)\mathbb {P} (V) \over {\mathbb {P} (+|V)\mathbb {P} (V)+\mathbb {P} (+|\sim V)\left[1-\mathbb {P} (V)\right]}}}$$

For any screening test, we must be cognizant of its sensitivity and specificity. The screening test has sensitivity ${\displaystyle \mathbb {P} (+|V)}$ and specificity ${\textstyle \mathbb {P} (-|\sim V)=1-\mathbb {P} (+|\sim V)}$. The sensitivity and specificity can be analyzed using concepts from the standard theory of statistical hypothesis testing:

• Sensitivity is equal to the statistical power ${\displaystyle 1-\beta }$, where ${\displaystyle \beta }$ is the type II error rate
• Specificity is equal to ${\displaystyle 1-\alpha }$, where ${\displaystyle \alpha }$ is the type I error rate

Therefore, the form of Bayes' theorem that is pertinent to us is:

$${\displaystyle \mathbb {P} (V|+)={(1-\beta )\mathbb {P} (V) \over {(1-\beta )\mathbb {P} (V)+\alpha \left[1-\mathbb {P} (V)\right]}}}$$

Suppose that we have developed a test with sensitivity and specificity of 99%, which is likely to be higher than most real-world tests. We can examine several scenarios to see how well this hypothetical test works:

• We screen welfare recipients for cocaine use. The base rate in the population is approximately 1.5%,[37] assuming no differences in use between welfare recipients and the general population.
• We screen men for the possibility of committing mass shootings or terrorist attacks. The base rate is assumed to be 0.01%.

With these base rates and the hypothetical values of sensitivity and specificity, we may calculate the posterior probability that a positive result indicates the individual will actually engage in each of the actions:

Even with very high sensitivity and specificity, the screening tests only return posterior probabilities of 60.1% and 0.98% respectively for each action. Under more realistic circumstances, it is likely that screening would prove even less useful than under these hypothetical conditions. The problem with any screening procedure for rare events is that it is very likely to be too imprecise, which will identify too many people of being at risk of engaging in some undesirable action.