What is the Bayes’ Theorem?
what is the Bayes’ Theorem?
In statistics and applied mathematics , the Bayes’ theorem also referred to as the Bayes’ rule may be a mathematical formula wont to determine the contingent probability of events. Essentially, the Bayes’ theorem describes the probability of an occasion supported prior knowledge of the conditions which may be relevant to the event.
The theorem is known as after english statistician, Bayes , who discovered the formula in 1763. it’s considered the inspiration of the special statistical inference approach called the Bayes’ inference.
Besides statistics, the Bayes’ theorem is additionally utilized in various disciplines, with medicine and pharmacology because the most notable examples. additionally , the theory is usually employed in several fields of finance. a number of the applications include but aren’t limited to, modeling the danger of lending money to borrowers or forecasting the probability of the success of an investment.
Intuitive understanding
The man was sitting together with his back to a wonderfully flat and perfectly square table. Then he asked his assistant to throw the ball on the table. Obviously, this ball could have landed and ended up but during a more analytical way.
So, he asked his assistant to throw another ball on the table and tell if it landed on the left or on the proper , or on the front or at the rear of the primary ball. He wrote that down, then he asked the assistant to throw more and more balls on the table.
He knows that with this method he could update his initial idea of where the primary ball was landed. But in fact , he could never be completely sure, but with each new proof, he would chop down the uncertainty and become more and more accurate.
And that’s how Bayes saw the planet , that’s his thinking experiment. isn’t “> it isn’t that he thought that the planet is not defined, that reality doesn’t exist, but that we will not realize it perfectly, and every one we will hope to try to to is to renew our understanding as more and more evidence emerges. i think it is a truly scientific-approach to knowledge.
Explanation
Above image we’ve two overlapped events A and B. It can be, for instance A — i buy wet today, B — it’ll be rainy today. In a method or another, many events are associated with one another , as in our example. Let’s calculate the probability of A as long as B has already happened.
Since B went on , the part which now matters for A is that the shaded part which is interestingly A ∩ B. So, the probability of A given B seems to be:
Therefore, we can write the formula for event B given A has already occurred by:
OR
Now, the second equation can be rewritten as:
In conclusion, that’s all that’s all that need to be drawn to return to the Bases theorem.Formula for Bayes’ Theorem
Where:
- P(A|B) — the probability of event A occurring, given event B has occurred.
- P(B|A) — the probability of event B occurring, given event A has occurred.
- P(A) — the probability of event A.
- P(B) — the probability of event B.
Example of Bayes’s Theorem
Assume you’re a securities analyst at an investment bank. consistent with your research of publicly-traded companies, 60% of the businesses that increased their share price by quite 5% within the last three years replaced their CEOs during the amount .
At an equivalent time, only 35% of the businesses that didn’t increase their share price by quite 5% within the same period replaced their CEOs. Knowing that the probability that the stock prices grow by quite 5% in 4%, find the probability that shares of a corporation that fires its CEO will increase by quite 5%.
Before finding the probabilities, you must first define the notation of the probabilities.
- P(A) — the probability that the stock price increases by 5%.
- P(B) — the probability that the CEO is replaced.
- P(A|B) — the probability of the stock price increases by 5% given that the CEO has been replaced.
- P(B|A) — the probability of the CEO replacement given the stock price has increased by 5%.
Using the Bayes’ theorem , we will find the specified probability:
Thus, the probability that the shares of a corporation that replaces its CEO will grow by quite 5% is 6.67% probability.
