Let e and f be events of sample space s of an experiment,then p(e / f) +p(ē /f) is?

Question

Answer

Answer

Definition

In probability theory, conditional probability is the probability of an event occurring given that another event has occurred. It is denoted as P(A/B), which means the probability of A happening given that B has already happened.

The complement of an event E is the event that E does not occur. It is denoted as E'.

Explanation

The probability of event E given that event F has already occurred is denoted as P(E/F). Similarly, the probability of the complement of event E given that event F has already occurred is denoted as P(E'/F).

Now, we know that the total probability of an event is equal to 1. Therefore, we can write:

P(E) + P(E') = 1

P(E/F) + P(E'/F) = 1

Subtracting the second equation from the first, we get:

P(E) - P(E/F) = P(E') - P(E'/F)

Adding and subtracting P(E' ∩ F) on both sides, we get:

P(E ∩ F) - P(E/F) = P(E' ∩ F) - P(E'/F)

P(E/F) - P(E' ∩ F) = P(E'/F) - P(E ∩ F)

Now, using the definition of conditional probability, we can write:

P(E ∩ F) = P(F) × P(E/F)

P(E' ∩ F) = P(F) × P(E'/F)

Substituting these values in the above equation, we get:

P(F) × (P(E/F) - P(E'/F)) = P(F) × (P(E'/F) - P(E ∩ F))

P(E/F) - P(E'/F) = P(E'/F) - P(E ∩ F)

Therefore, we can conclude that:

P(E/F) - P(E'/F) = P(E'/F) - P(E ∩ F)

P(E/F) + P(E'/F) = P(E ∩ F) + P(E'/F)

Hence, we can say that the probability of event E given that event F has occurred plus the probability of the complement of event E given that event F has occurred is equal to the sum of the probabilities of the intersection of events E and F with the complement of event E given that event F has occurred and the intersection of events E' and F given that event E has occurred.

Answer

P(e/f)

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