# 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.