Question

State and prove the multiplication theorem of probabilities for two independent events:

Answer 1

Statement: Let A and B be any two independent events with respective probabilities P(A) and P(B). Then, the probability of occurrence of events A and B is P(A∩B) = P(A). P(B) 

Proof: Suppose, a random experiment, results ‘n’ outcomes, of which’m’, outcomes are favorable k to events A and another random experiment results n₂ outcomes of which m₂ outcomes are favorable to event B.

The occurrence of the events A and B together m₁ and m₂ i.e., m₁ × m₂ favorable events out of n₁, and n₂ i.e., n₁ × n₂ outcomes. 

∴ Probability of ocurrence of events A and Btogther is.

Hence the proof.