∙Except by fluke, it is not true thatPB|APB|Ac1. [Bycontrast, it is true thatPB|APBc|A1 always.]∙To get Bayes’ rule, we use the previous expression forPB:PA|BPB|A PAPB|A PAPB|Ac PAc∙In other words, we can obtainPA|Bby knowingPB|A,PB|Ac,andPA.64
EXAMPLE: Deciding whether to locate a business at a particular site.Want to know the chances that the business will be profitable. Canconduct a study that provides input.Asite is profitableBstudy says site is profitable65
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∙Suppose 32% of all similar sites are profitable, soPA.32. Basedon past experience where studies were conducted, also knowPB|A.65 andPB|Ac.29. Using Bayes’ rule,PA|B.22.214.171.124.29.68≈.513∙In other words, if the study says the site will be profitable, the siteactually will be profitable with probability .513. This is notably higherthan the unconditional probability – that is, the probability withoutknowing the outcome of the study – which is .32.∙What isPA|Bc?66