Total Probability

Total Probability is the sum of all possible outcomes from an event, and it can be calculated by adding up the probabilities of each outcome. For example, if you roll two dice, there are six possible outcomes: 1-2, 3-4, 5-6. The total probability for this roll would be 6/6 or 1.

The rule of total probability is used to determine the probability of an event A, when it’s hard to determine A’s probabilities to calculate it directly. Instead, you use the likelihood of a related occurrence B, to compute the chance of A. Total probability divides probability calculations into distinct sections.

Total Probability Theorem

Total Probability Theorem is a mathematical theorem that deals with the probability of a random event. It is used to calculate the probability of an event happening, given that certain conditions are met.

The Total Probability Theorem is a mathematical theorem that shows that if you know the probabilities of all possible outcomes, you can calculate the probability of any outcome by multiplying these probabilities together.

The following is the expression of the Total Probability Theorem for your total probability study material:

‘If A1, A2, A3,……are the partitions of the sample space S such that the probability of none of these events is equal to zero, then the probability of an event ‘E’ occurring in such a sample space is given as’, P(E) = ΣP(E|Ai)*P(Ai)

The Law of Total Probability

The law of total probability is a mathematical principle that states that the likelihood of an event occurring equals the sum of all possible outcomes divided by the total number of possibilities.

The Law of Total Probability was first proposed by the French mathematician Pierre-Simon Laplace in 1812. This law states that “the probability of a single event is equal to the number of cases multiplied by the probability of each case.”

The law is essential in understanding how chance and randomness work. It can help us understand why we do not have control over events. The law of total probability is a fundamental principle in probability study. It states that the probability of a particular outcome is equal to the sum of all possible outcomes. This means that if there are six different outcomes, each with a 1/6 chance, then there are six different ways for this event to occur, and so we multiply these probabilities together and get 1/36 or 3%.

The law of total probability can be expressed via the following formula that you should know for your study material notes on Total Probability:

P(A) = P(A∩B) + P(A∩Bc)

Probability Tree

A probability tree is a decision-making tool used to help people make decisions by considering the various outcomes and probabilities.

• The process starts with drawing a tree on a piece of paper.
• Next, we assign some values to each branch, and then we calculate the total probability for each branch.

To find the total probability for a given event, we need to know the possible outcomes. Our example has four possible outcomes – success, failure, success with one loss and failure with one win. A sum of these four probabilities represents the total probability.

Probability of Random Events

In this section, we will discuss the probability of a random event. An unexpected event is the occurrence of an unpredictable outcome of an experiment or activity.

The total probability of a random event is the sum of all possible outcomes in an experiment or activity. There are two ways to calculate the total probability:

• Total Probability formula- The sum of all possible events divided by the number of events.
• Expected value formula- The average value over all possible outcomes.

Probability of Mutually Exclusive Events

When two incidents cannot happen at the exact moment, they are mutually exclusive. The consequence of mutually exclusive occurrences is always different. When one of these occurrences occurs, it prohibits the second event. For instance, if you get a “Heads” on the coin toss, you won’t get a “Tails.” These are two events that can’t happen at the same time.

You may argue that almost any two outcomes are mutually incompatible at this point! No, not at all. Consider the case of two coins being tossed at the very exact moment. The chance of either a Heads or a Tails on one of these coins is unaffected by the occurrence of H or T on the other.

Conclusion

Total probability is a mathematical concept used to determine the probability of a particular event happening.

Total probability is defined as the sum of all possible outcomes. It can be calculated by adding up all the different possible outcomes of an event and dividing that number by the total number of possible outcomes.

The formula for calculating the total probability for your study material notes on Total Probability:

P(A) = P(A∩B) + P(A∩Bc)