Multiplication Theorem of Probability

How do we decide what will happen or how many ways a particular thing could occur? We decide on stances that become halves of the happenings, whether it will happen or not (essential being 50 – 50). Now, where does this 50 – 50 come from? In Greek mythology, it is stated that there are as many possibilities of a stance as much as there are the items. From there comes possibility, and hence the derived word “Probability.” Probability is a branch in mathematics that looks over the happenings of random or unknown events, or we can the branch of mathematics which deals with the numerical description of the extent of the occurrence of how likely a random event like tossing a coin, selecting a card from a deck, rolling dice, etc. can occur is called probability. The probability of any event lies between 0 to 1. Probability means the ratio of the likely outcomes to the total number of possible outcomes. The meaning of probability is a possibility. After this comes a term, Multiplication Theorem in probability. To understand, one needs a vast imagination strength to understand and re-educate situations we are dealing with. 

Understanding The Theorem

The multiplication theorem of probability is a conditional probability, where we look for the condition to occur and get a stigmatized order of their occurrence and non-occurrence. In the Multiplication theorem of probability, some other events are affected when some of the events occur. When a particular event A has occurred, and we need to find the occurrence probability of the same, we will take the occurrence probability of the given event A rather than looking up for some unknown event to attain value outcomes. So, this mathematical theorem of probability states that the “probability of the simultaneously occurring two events A and B will be equal to the product of one of these events and the conditional probability of the other one, provided that the first event has occurred.

Simplicity Of Probability

Probability plays a much-textured role in planning out decisions in each aspect of our day-to-day lives and talking about the Multiplication theorem of probability. Let’s make it easy for you to understand it more briefly. Let’s assume you are playing monopoly with your friends, and monopoly requires the use of two dice. So when one of your friends throws dice, he hides the result, and he says that by adding on two of these two numbers, you will get an even valued number. Does this lead to many options in your mind of what exactly those two numbers will be? This was the basic concept overview of the Multiplication theorem of probability in theoretical words.

The formula

Moving on further into some mathematical terms and conditions of the multiplication theorem of probability. The mathematical representation or the basic formula for the same is –                

P (A ∩ B) =P(A)P(B|A).                       

The multiplication theorem becomes easier when additional information is provided with the unknown event. Below is one of many examples of a probability formula that will help you easily understand it with interest!                   

Let’s take the same example of playing monopoly where your friend had thrown the dice and didn’t expose the outcomes. So, the possible outcomes of throwing two dices would be-                                      

S= {(x, y): x, y = 1,2,3,4,5,6}

There would be 36 sample spaces (as there are two dices and six outcomes for each), the possibility/probability of happening of any outcome would be:

P (E1) =1/36                                     

We don’t know the result as our friend did not disclose it to us, but the information provided alongside some of the numbers on the dice is even.                                          

 So, the sum of the actual outcomes would be:

A= {(1,1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)}                                

This means out of 36 outcomes, 18 would be the even ones. So, the probability of these outcomes be:                                                   

P (A/E1) = 18/36 = 1/2                                                   

The multiplication theorem of probability helps us determine miscellaneous outcomes of any given event.

Conclusion: 

Probability is a branch in mathematics that looks over the happenings of random or unknown events. The probability of any event lies between 0 to 1. The mathematical theorem of probability states that the “probability of the simultaneously occurring two events A and B will be equal to the product of one of these events and the conditional probability of the other one, provided that the first event has occurred. It is a conditional probability, where we look for the condition to occur and get a stigmatized order of their occurrence and non-occurrence.