Mathematician 30 September 2022

Over the past week, I had a chance to review things that were the fundamentals of the theory of probability, and the Bayesian Probability is one of the popular subjects that has been adapted in many ways. The content in general is nothing complex, but it can still be applied to many things to help with the decision-making.

Actually, the concept of the currently popular predictive model is also explainable with the Bayesian Theorem where the model is set to predict things “given” historical data. The translated formulas will then explain the predictive model concept quite clearly, but it will not be mentioned in this article yet.

In addition, the Bayesian Theorem can help us choose to receive news and information better. For example, when we learn about the infection rate of a communicable disease and its fatality rate, we can then use the Bayesian Theorem to calculate how fearsome the disease is by seeking a little more information to help support the daily decision-making.

The Bayesian Probability talks about the conditional likelihood of an event based on the occurrence of another event, or the so-called a “given”. If you have studied this, it is written as P(A|B) which reads as how often A happens given that B happens.

Let’s assume that I am attending a friend’s birthday party which has 20 guests, and there is a lucky draw for 5 cruise tickets for the guests.

The host then puts the tickets into 2 boxes. We will call them box X and box Y. Therefore, the probability is summarized below:

- Event A is choosing the boxes in which the element is A = {box X, box Y}
- Event B is drawing the tickets in which the element is B = {cruise tickets won, cruse tickets not won}
- The probability of choosing box X and box Y is ½ and ½ respectively.
- The probability of winning and not winning the cruise tickets is 5/20 and 15/20 respectively.
- Written as P(A= box X) =1/2 or P(B= cruise tickets not won)=15/20, for instance

After the lucky draw, it turns out I didn’t win. So, at the end of the party, I talk to my friend who’s the host. I find out that box X and box Y had different numbers of tickets. In box X, there were 3 cruise tickets out of 10 lucky draw tickets, and box Y had 2 out of 10 tickets. Based on this information, it can be concluded as follows:

- The probability of winning the cruise tickets choosing box X is P(B= winning the cruise tickets | A=box X) = 3/10.
- The probability of winning the cruise tickets choosing box X is P(B= not winning the cruise tickets | A=box X) = 7/10.
- The probability of winning the cruise tickets choosing box Y is P(B= winning the cruise tickets | A=box X) = 2/10.
- The probability of winning the cruise tickets choosing box Y is P(B= not winning the cruise tickets | A=box X) = 8/10.

If we summarize the overall probability, we can then divide it into different scenarios using the joint probability of A and B events as follows:

- Winning the cruise tickets from box X: P (B=winning the cruise tickets, A=box X) = 3/20
- Not winning the cruise tickets from box X: P (B=not winning the cruise tickets, A=box X) = 7/20
- Winning the cruise tickets from box Y: P (B=winning the cruise tickets, A=box Y) = 2/20
- Winning the cruise tickets from box X: P (B=not winning the cruise tickets, A=box Y) = 8/20

You can see that the joint probability is represented by “,” (comma) which is different than the conditional probability that uses “|” (given).

Based on all the scenarios, Bayes has summed up the relationships as follows:

- P(A, B) = P(A|B) x P(B)
- P(B, A) = P(B|A) x P(A)
- When P(A, B) = P(B, A), it is:

P(A|B) x P(B) = P(A, B) = P(B|A) x P(A)

And when P(A|B) x P(B) = P(B|A) x P(A), switching sides of the equation will get:

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

When you have made it here, many of you may wonder what use we can make of it, and why we need to know about it.

Let’s get back to the birthday party in the beginning. After learning that a friend has won a cruise ticket from the lucky draw, I wonder which box he drew from. The host then challenged me to guess it. There’s another cruise ticket. If I guess right, I get the ticket. So, I have to make an educated guess this time. Considering the Bayesian Probability, it can then be concluded as follows:

*P(A=box X|B=winning the cruise ticket)
= P(B=winning the cruise ticket | A=box X) x P(A=box X) / P(B=winning the cruise ticket)
= (3/10 x 1/2) / 5/20 = 3/5*

And

= P(B=winning the cruise ticket | A=box Y) x P(A=box X) / P(B=winning the cruise ticket)

= (2/10 x 1/2) / 5/20 = 2/5

From this example, I would have to choose box X because the probability is greater than box Y (3/5 > 2/5).

In fact, I don’t need to calculate the probability to make an easy choice because I know that box X has more tickets than box Y. Naturally, the probability that the person who won drew the ticket from box X is greater. In the end, it doesn’t matter if I guess correctly or not because I can only calculate the “probability”, but I consider it as a decision made based on the most basic information I had.

In reality, the Bayesian Theory can also be applied to more complex and important problems.

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