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Have you had a reason to wish you could predict with near accuracy what the future holds?

It’s possible to predict with a high level of confidence what the future has in store based on prior information. You could use Bayes theorem or Bayes rule to determine the conditional probability of events.

Note that a conditional probability is the likelihood of an outcome occurring, based on a previous outcome.

*What is Bayes theorem or Bayes rule?*

Bayes theorem or Bayes rule is a mathematical formula for determining the conditional probability of events. It is useful in calculating the probability of a first stage event given that a second event occurred.

Bayes theorem is a posterior probability that an event will happen after an event or background information has been considered. In other words, this probability is based on prior knowledge of the condition that might be fundamental to the event.

Rev. Thomas Bayes developed two formulas. These are;

P(A/B) = P(B/A) * P(A)/P(B)

Where;

A and B = events,

P(A/B) = probability of A given B is true,

P(B/A) = probability of B given A is true,

P(A) = the probability of event A

P(B) = the probability of event B

P(B) = P(B/A) * P(A) + P(B/A’) * P(A’)

Events A and B are independent of each other.

Or

P(A/B) = P(B/A) * P(A)/P(B/A) * P(A) + P(B/A’) * P(A’)

Where P(A’) = 1 – P(A)

The formula can be extended to three or more mutually exclusive events.

Let us consider an example:

If it is estimated that there is 30 percent chance that the level of unemployment will rise by 2 percent next year. If this occurs there’s a 80 percent chance that government will come up with a social welfare program. If the level of employment doesn’t rise, the probability of government funding such a program is 40 percent. If this program was funded by government, what is the probability that the level of employment rose by more than 2 percent?

Applying the Bayes theorem formula;

P(A) = 0.3

P(A’) = 0.7

P(B/A) = 0.8

P(B/A’) = 0.4

Therefore by Bayes theorem,

```
P(A/B) = P(B/A) * P(A)/P(B/A) * P(A) + P(B/A’) * P(A’)
```

Therefore,

P(A/B) = 0.8 * 0.3/ 0.8 * 0.3 + 0.4 * 0.7

```
= 0.46.
```

Therefore the probability of A given B is 0.46 and is less than 1. This means events A and B are mutually exclusive (they are independent).

*Why is b Bayes theorem important for finance and business?*

Bayes theorem is important for finance and business in the following ways;

- It is used to calculate changes in interest rate on borrowed money. Interest rates may increase or decrease as a result of central banks’ monetary policy intervention in the money market.

2. It’s a fact that monetary policy has significant impact on the money supply in an economy which in turn, affects interest rates. Bayes rule uses information about a country’s monetary policy to determine changes in interest rate on borrowed money

3. It is also used to determine the direction of foreign exchange rates. Factors such as terms of trade, political stability, interest rates, government debt and others are used as information to determine changes in currency exchange rates using Bayes rule.

4. Bayes rule or Bayes law can be used to assess the impact of government policies on net income. Apart from government policies, Bayes theorem can be used to determine the impact of other external factors on net income. Such uncontrollable events may include climatic changes, litigations, economic shifts (growth or decline), technological innovations and others.

5. Bayes rule can also be used to determine credit worthiness.

In conclusion, Bayes theorem is an important part of critical thinking in business decision making. It should help a business that applies it to make best business decisions..

***

*Below are what others think;*

In investing, the concept of conditional probability is an important one. During 2020, there were certain investments which in theory were too risky without one knowing how the virus would turn out. As it became clear that the virus, while tragic, was not an insurmountable hurdle, certain names became investable. Some of these names would later go on to double and triple. And then when the vaccines came out, another set of names became investible. Most people invest based on biases, but if you can objectively estimate new probabilities (and therefore expected value of outcomes) on a real-time basis, you can be more opportunistic than peers and improve your chances in the public markets.

Stock Market example:

Suppose that you study a specific aspect of the stock market (e.g. utility futures, precious metals, technology stocks, agriculture futures …) and determine the following:

When a certain situation occurs, your superior insight will (in the long run) produce a profit, after broker’s fees.

To ensure a profit, you instruct your broker that if the transaction can not be made within (1/2)(1/2) point of the opening price that morning, the transaction should be cancelled.

Over time, despite the edge that your insight has given you, your transactions may result in an overall loss. This is because the transactions where your analysis is accurate, are more likely than not to be recognized by others as favorable transactions.

This implies that it is plausible that the transactions that are the most profitable, you will be denied, because a disproportionate amount of the time, for these particular transactions, the stock price will have moved too far against you.

Therefore, you make only those transactions where the stock price has changed no more than (1/2)(1/2) point. In a disproportionate number of these transactions, either your analysis was flawed, or there is *inside information* unavailable to you which will prevent the transactions from being *that* profitable.

This is a clear case of Bayes Theorem.

Let AA be the event that the transaction will be profitable.

Let BB be the event that the stock price has not significantly moved against you.

For simplicity, assume that event BB will occur as follows:

- (1/2)(1/2) the time, your analysis + information is good, and you are able to make the transaction because others were slow.
- (1/2)(1/2) the time the transaction will not be profitable re analysis mistake or unknown information.

Then p(A|B)=p(AB)p(B)p(A|B)=p(AB)p(B).

It is very plausible, in such a scenario that

p(AB)p(B) is significantly less than p(A).

.