That's just the usual equation with a restriction to ##E## as the universal set or sample space. Given the proviso, as above, that the ##A_i## (when restricted to ##E##) partition ##E##.red65 said:
The law of total probability with extra conditioning is a mathematical concept used in probability theory. It states that the probability of an event is equal to the sum of the probabilities of that event occurring under different conditions or scenarios.
The regular law of total probability only considers one conditioning event, while the law of total probability with extra conditioning takes into account multiple conditioning events. This allows for a more comprehensive understanding of the probabilities involved.
For example, if we want to calculate the probability of getting a red card in a deck of cards, we can condition this probability on the suit of the card. The law of total probability with extra conditioning would then take into account the probabilities of getting a red card in each suit (hearts, diamonds, clubs, and spades).
The law of total probability with extra conditioning is used in various fields such as statistics, economics, and engineering. It can help in decision making by providing a more accurate understanding of the probabilities involved in a given scenario.
One limitation is that it assumes that the conditioning events are independent of each other, which may not always be the case in real life. It also requires accurate and complete data to be effective in its application.