The COVID-19 pan­dem­ic was a sad but per­fect oppor­tu­ni­ty to engage with math­e­mat­ics. Sud­den­ly, we were all con­front­ed with inci­dence rates, over­whelmed with sta­tis­tics, and had to deal with prob­a­bil­i­ties. On a very per­son­al lev­el, with the like­li­hood of get­ting infect­ed or the ques­tion of how reli­able the new­ly devel­oped COVID-19 tests were. Ques­tions that could decide on free­dom, health, and for some even on life and death.

With­out math­e­mat­i­cal knowl­edge, how­ev­er, none of these ques­tions can be prop­er­ly answered. Here is a con­crete example:

«Accord­ing to the man­u­fac­tur­er, a COVID-19 rapid test offers a 95 per­cent prob­a­bil­i­ty of cor­rect­ly detect­ing a COVID-19 infec­tion. Con­verse­ly, the test false­ly indi­cates a pos­i­tive result with a prob­a­bil­i­ty of 2 per­cent. Now let’s assume you test pos­i­tive while COVID-19 cur­rent­ly has a preva­lence (pro­por­tion of peo­ple in the pop­u­la­tion who are infect­ed at that time) of one per­cent (a val­ue quite real­is­tic for the peaks of the COVID-19 waves): What do you think is the prob­a­bil­i­ty that the test is correct?»

At first glance and intu­itive­ly, many peo­ple would prob­a­bly con­sid­er such a test quite reli­able. Math­e­mat­i­cal­ly speak­ing, how­ev­er, it is not — as the cal­cu­la­tion using the so-called Bayes’ the­o­rem shows, used to cal­cu­late the prob­a­bil­i­ty of an event occur­ring based on pri­or knowl­edge of relat­ed con­di­tions or events. It states:

P(A|B) = (P(B|A) * P(A)) / P(B)

In our case:

• A rep­re­sents the prob­a­bil­i­ty that you are actu­al­ly infect­ed with COVID-19.
• B rep­re­sents the prob­a­bil­i­ty that the test comes back positive.

We are look­ing for P(A|B), the prob­a­bil­i­ty of actu­al­ly hav­ing COVID-19 if the test is pos­i­tive. Here are the nec­es­sary probabilities:

• P(A): The preva­lence of COVID-19 in the pop­u­la­tion is 1%, so P(A) = 0.01.
• P(B|A): The prob­a­bil­i­ty that the test is pos­i­tive if you actu­al­ly have COVID-19 is 95%, so P(B|A) = 0.95.
• P(B): The prob­a­bil­i­ty that the test comes back pos­i­tive, regard­less of whether you are infect­ed or not. To cal­cu­late this, we need to add the prob­a­bil­i­ties for false pos­i­tive and cor­rect pos­i­tive results:
• P(B) = P(B|A) * P(A) + P(B|¬A) * P(¬A)
• P(B|¬A) is the prob­a­bil­i­ty that the test comes back pos­i­tive even though you don’t have COVID-19 (false pos­i­tive). This is 2%, so P(B|¬A) = 0.02.
• P(¬A) is the prob­a­bil­i­ty that you don’t have COVID-19. Since the preva­lence is 1%, the prob­a­bil­i­ty of not being infect­ed is 99%, so P(¬A) = 0.99.
• P(B) = 0.95 * 0.01 + 0.02 * 0.99 = 0.0095 + 0.0198 = 0.0293

Now we can apply Bayes’ the­o­rem to cal­cu­late P(A|B):

P(A|B) = (P(B|A) * P(A)) / P(B) = (0.95 * 0.01) / 0.0293 ≈ 0.3242

Accord­ing to Bayes’ the­o­rem, the prob­a­bil­i­ty that the test is cor­rect and you actu­al­ly have COVID-19 is about 32.42 per­cent. A sur­pris­ing­ly low val­ue for many.

Con­front­ed with media infor­ma­tion on a dai­ly basis
Cal­cu­la­tions like these demon­strate the impor­tance of a basic under­stand­ing of math­e­mat­ics and con­di­tion­al prob­a­bil­i­ties for essen­tial ques­tions in our every­day lives — and for a mul­ti­tude of deci­sions. Bayes’ the­o­rem, devel­oped in the 18th cen­tu­ry by the Eng­lish math­e­mati­cian, sta­tis­ti­cian, philoso­pher, and Pres­by­ter­ian min­is­ter Thomas Bayes, remains a cru­cial foun­da­tion to this day. It plays a sig­nif­i­cant role in data min­ing, spam detec­tion, arti­fi­cial intel­li­gence devel­op­ment, or qual­i­ty man­age­ment. Last but not least, draw­ing con­clu­sions from data, as Christoph Wass­ner, Ste­fan Krauss, and Lau­ra Mar­tignon empha­sized in a 2002 paper, forms the basis for many pro­fes­sions: in med­ical diag­noses, judi­cial judg­ments, and eco­nom­ic deci­sions. «The same applies to the ‹informed cit­i­zen› who is con­front­ed with media infor­ma­tion on a dai­ly basis.»

Some every­day exam­ples may illus­trate the prac­ti­cal appli­ca­tions of Bayes’ the­o­rem in our lives. For instance, when choos­ing between dif­fer­ent trans­porta­tion options for our dai­ly com­mute, we can use Bayes’ the­o­rem to cal­cu­late the like­li­hood of delays based on past expe­ri­ences, traf­fic con­di­tions, and sea­son­al fac­tors. This enables us to make more informed deci­sions about which mode of trans­porta­tion offers the high­est prob­a­bil­i­ty of arriv­ing at work on time.

Sim­i­lar­ly, Bayes’ the­o­rem can assist us in mak­ing pur­chase deci­sions by incor­po­rat­ing addi­tion­al infor­ma­tion, such as the num­ber of reviews or our spe­cif­ic require­ments, to esti­mate our sat­is­fac­tion with a prod­uct. More­over, mete­o­rol­o­gists employ this the­o­rem to pre­dict weath­er events like rain, based on cur­rent con­di­tions and his­tor­i­cal data, allow­ing us to plan our dai­ly activ­i­ties more effec­tive­ly. By apply­ing these fun­da­men­tal math­e­mat­i­cal prin­ci­ples, we can make more informed and ratio­nal deci­sions in var­i­ous aspects of our lives. So, per­haps it’s time to re-engage with this for­mu­la that has been devel­oped more than 250 years ago, but nev­er gets old.