Joint Probability confusion

Hey guys,

There’s the concept of Joint Probability which confuses me in separate context, allow me to explain:

  1. Joint Probability and Multiplication Rule
  • Probability that both A and B will occur is their joint probability: P(AB)
  • P(AB) = P(A|B) x P(B)
  1. Addition Rule for Probabilities
  • The probability that A or B occurs, or both occurs, is equal to:
  • P(A or B) = P(A) + P(B) - P(AB)
  • In an example in the curriculum, there was the following question: Firm X has issued two callable bonds with a maturity of 2 and 5 years respectively. Probability that Bond A will be called is 0.6 and probability that Bond B will be called is 0.5. The probability that at least one of the bonds will be called is closest to?
  • P(A or B) = P(A) + P(B) - P(A and B)
  • P(A or B) = 0.6 + 0.5 - (0.6 x 0.5)

Realize how we just did 0.6 x 0.5 to fulfill P(AB) but the prior formula says that P(AB) = P(A|B) x P(B), but in the example they just assumed P(AB) = P(A) x P(B).

What is going on?

A and B are independent events.

When the events probabilities are independents P(AB) = P (A|B) x P(B).

This is because if event B happens don’t change the probability of event A.

P(A) = P(A|B) only when A and B are independent.

What worked for me in quantitative methods was using a statistical book I used in college and study from the book, absolutely it was explained more in detail.

Please, let me know if you did understand. You can write me by mail o quantitative methods question, I really did very well in the review and would be good for me to refresh the topics explaining.

Here, events are independent, so P(AB) = P(A)P(B) is used.