The Difference Between Logical Probability and Subjective Probability Using a Simple Example (Involving Cupcakes)

Mathematicians have been trying to answer them for decades now and the experiments have led to multiple interpretations of what it means to say something is likely to occur.In this article we illustrate two ways of interpreting probabilities with a very simple example. Each makes sense individually but are very different streams of thought. For example, let us consider our novice cook Anna who gives us home cooked brunch every Sunday. Anna bakes either cupcakes or brownies but never both as dessert. Therefore, dessert selection is mutually exclusive and collectively exhaustive.Logical ProbabilityConsider the Hypothesis H.According to logical probability H must be supported to a high degree of force by evidence E. It is denoted as c(H, E).The high degree of evidence is important to note as it reinforces the condition that logical probability is believed to be free from subjectivity. In this case, unlike classical probability, when we think about a set of outcomes of an event, we can assign multiple values of probability to multiple outcomes.As per the information I have, Anna had a very busy week. Anna also finds it easier to bake brownies than cupcakes when she has had a busy week. Whenever Anna has a busy week, she bakes brownies 60% of the time. An additional information is that I like brownies more than cupcakes.My hypothesis H is, ‘I will have brownies’.My evidence E is, ‘Anna had a very busy week and that Anna finds it easier to bake brownies than cupcakes. Whenever Anna has a busy week, she bakes brownies 60% of the time’.H is probably true considering E supports H to a high degree.Therefore, after considering the facts, I feel that probability of me having brownies is over a 70% – according to the logical probability.I go over to Anna’s house and find that she made cupcakes.How did my logic probability fail me?For the answer, I need to examine the strength of my evidence ‘E’. Had I been truly objective as I was supposed to be? Did I have enough data to prove that ‘whenever she had a busy week, she baked brownies 60% of the time’ part of the evidence? I correlated ‘having a very busy week’ and ‘she finds brownies easier to bake’ and made the evidence stronger. It added subjectivity to the incomplete evidence. Also, the fact that I liked brownies more than cupcakes added more subjectivity to the evidence.Therefore, the fault with logical probability is that; one we might not always have the complete evidence. And two, interpretation of the evidence plays a role in the addition of subjectivity to it.Subjective Probability or Bayesian ProbabilityIn the previous section of logical probability, we saw how subjectivity crept into our so-called objective evidence. Subjectivists or Bayesians embraces this subjectivity. Subjectivists or Bayesians (See our article on Bayesian Reasoning for a detailed explanation) regards probability as ‘degree of belief’ or to what extent the person assigning the probability to an event believes it to be true or false.However, Bayesians don’t just use their subjective degree of belief. They use evidence from the past, called prior and the evidence or data we have to find the conditional probability. The resulting probability is called posterior.The process does not stop there, for the next iteration or to update the accuracy of the posterior, we use it as the prior for finding the new posterior. This process of updating is iterated.Now let us come back to our brunch. We get some information from the baker herself so I can find out the probability of getting my favorite dessert.Anna says out of all the desserts she baked, about 70% are brownies and 30% must be cupcakes. When she has had a busy week, she finds it easier to bake brownies and does so about 80% of the time.With this information in hand, how do we find the probability of me having brownies this Sunday?P(Baking Brownies|Busy Week) =? P(Baking Brownies) = 0.7 (prior)P(Busy Week|Baking Brownies) = 0.8 (evidence)P(Baking Brownies|Busy Week) =  (0.8 * 0.7 ) / ((0.7*0.8)  + (0.3*0.2))= 0.56/1.06= 0.35 (posterior)Therefore, even though as per the information it looks as if the odds that Anna might have baked brownies is good, by using conditional probability, we see that it is only 35%.Next week, before we go Anna’s, we go Anna’s we do the same calculation. We will use 35% as the prior instead of 70%.There will be a certain degree of variation of probability from each prior and posterior.It is obvious there are lots of room in this area for further research as all these methods have their own caveats. Each concept has its own unique utility but a there is no unified interpretation yet that would cover all dimensions of probability yet.As we resign to this fact I quote Jacob Bernoulli who said, “It is utterly impossible that a mathematical formula should make the future known to us, and those who think it can once believed in witchcraft”.E-mail us at to inspire our readers with your story – be it your success story or a lesson learned, share what you learned or send some love to a friend. We would love to hear from you!