Theory on Bayes’ Theorem – Simple Problems

Bayes’ theorem is a mathematical formula that helps us calculate the probability of an occurrence happening, based on our current knowledge of other events. It can be used in a variety of different situations, from predicting the weather to calculating how likely it is that a criminal will re-offend. In this blog post, we will discuss some simple problems that can be solved using Bayes’ theorem. Stay tuned for more posts on more complex applications of this theorem!

What is Bayes theorem?

Bayes theorem is a mathematical formula used to calculate the probability of an occasion occurring, given that another event has already occurred. That is, the Bayes theorem is basically a rule of thumb that helps us update our beliefs in the light of new evidence.

The theorem is named after English statistician Thomas Bayes (1701-1761), who first formulated it in a letter to Richard Price in 1763.

Bayes theorem and Conditional Probability:

Bayes theorem is a way of calculating the probability of an occurrence, given information about conditions that might be related to the event.

For example, imagine you’re trying to figure out the probability that it will rain tomorrow. You might start by looking at the forecast, which might give you a 60% chance of rain. But there are other factors that can affect the probability of rain, such as the temperature and the amount of humidity in the air.

Bayes theorem can help you take all of these factors into account and calculate a more accurate probability of rain.

What is the Bayes theorem formula?

Here’s the formula for the Bayes theorem:

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

P(A|B) is the probability of A given B.

P(B|A) is the probability of B given A.

P(A) is the probability of A.

P(B) is the probability of B.

In the above example, A is the independent variable and B is the dependent variable.

In simple terms, the Bayes theorem formula is a way of calculating the probability of an occurrence ensuing, given that another event has already occurred.

Now let us find out the Bayes theorem probability formula if the A is the binary variable.

P(A|B)= P(B|A)P(A)P(B)

where A and B are the events.

 P(A|B)= Probability of A provided B is true.

P(B|A) =Probability of B  provided A is true.

P(A), P(B)= independent probabilities of A and B.

Now, we can use this Bayes theorem probability formula to calculate the probability of an occasion occurring.

Examples of Bayes theorem:

Simple Example: Let’s say we have a jar of 200 marbles. 100 of them are black, and the rest are red. If we randomly pick a marble out of the jar, what is the probability that it is black?

By definition, the probability of an occurrence happening is the number of ways that an event can occur divided by the total number of possible outcomes. In this case, there are 100 black marbles and 200 total marbles, so the probability of picking a black marble is 100/200, or 50%.

Now, let’s say we have two jars of marbles.Jar 1 contains 110 black marbles and 30 red marbles. Jar 2 contains 20 black marbles and 80 red marbles. If we randomly pick a jar and then pick a marble out of that jar, what is the probability that the marble is black?

There are two possible scenarios: either we pick Jar 1 or we pick Jar 2 . If we pick Jar 1, then the probability of picking a black marble is 110/140, or 79%. If we pick Jar Jar two, the probability of picking a black marble is 20/100, or 20%. Since both scenarios are equally likely, we need to take a weighted average of the two probabilities. This gives us a probability of black of (79% x 0.50) + (20% x 0.50), or 49.50%. So, even though the probability of picking a black marble out of anyone’s jar is lower than 50%, the probability of picking a black marble overall is just under 50%.

This is an example of the law of total probability, which says that the probability of an occurrence happening is the sum of the probabilities of all the ways that an event can happen. The law of total probability is closely related to Bayes’ theorem. In fact, you can use the law of total probability to derive Bayes’ theorem.

Conclusion

The Bayes theorem is a powerful tool that can be used to solve many probability problems. Bayes theorem is important because it allows us to calculate the probability of something happening, even if we don’t have all the information. For example, imagine you’re trying to calculate the probability of it raining tomorrow. You might not know what the weather is going to be like, but you can use Bayes theorem to calculate the probability based on past data.