Lesson 5: Independent Events

In this lesson, we learn what it means for two (or more) events to be independent. We'll formally learn, for example, why we say that the outcome of the flip of a fair coin is independent of the flips that came before it. A couple plans to have three children. What is the probability that the second child is a girl? And, what is the probability that the second child is a girl given that the first child is a girl? This example leads us to a formal definition of independent events. Events 𝐴 and 𝐵 are independent events if the occurrence of one of them does not affect the probability of the occurrence of the other. That is, two events are independent if either: 𝑃 ( 𝐵 | 𝐴 ) = 𝑃 ( 𝐵 ) (provided that 𝑃 ( 𝐴 ) > 0 ) or: 𝑃 ( 𝐴 | 𝐵 = 𝑃 ( 𝐴 ) ) (provided that 𝑃 ( 𝐵 ) > 0 ). Now, since independence tells us that 𝑃 ( 𝐵 | 𝐴 ) = 𝑃 ( 𝐵 ) , we can substitute 𝑃 ( 𝐵 ) in for 𝑃 ( 𝐵 | 𝐴 ) in the formula given to us by the multiplication rule: 𝑃 ( 𝐴 ∩ 𝐵 ) = 𝑃 ( 𝐴 )

In this lesson, we learn what it means for two (or more) events to be independent. We'll formally learn, for example, why we say that the outcome of the flip of a fair coin is independent of the flips that came before it. A couple plans to have three children. What is the probability that the second child is a girl? And, what is the probability that the second child is a girl given that the first child is a girl? This example leads us to a formal definition of independent events. Events 𝐴 and 𝐵 are independent events if the occurrence of one of them does not affect the probability of the occ