A probability like “the probability that a randomly selected household has a landline given that the household has a cell phone” is known as a conditional probability. There's even a special notation for it. In this example, we could write P(landline | cell phone) where the line is read as given.
Conditional probability is the probability that one event happens given that another event is already known to have happened is called a conditional probability. Suppose we know that event A has happened. Then the probability that event B happens given that event A has happened is denoted by P( B | A ). In conditional probability, order matters––just like in algebraic order of operations
When you’re looking at conditional distributions in a two-way table, it can be helpful to cover up the rows or columns that are not part of the “given.”
Calculating Conditional Probabilities
To find the probability P (A | B ), use the formula P( A | B ) = P( A & B ) / P(B)
The conditional probability P (B | A) is given by P( B | A ) = P( B & A ) / P(A)
Another way to look at it is this: P(both events occur) / P(given event occurs)
General Multiplication Rule
The probability that events A and B both occur can be found using the general multiplication rule P(A & B) = P(A) * P(B | A) where P(B|A) is the conditional probability that event B occurs given that event A has already occurred.
Two events A and B are said to be independent if the occurrence of one event does not change the probability that the other event will happen. In other words, events A and B are independent if P(A|B) = P(A) and P(B|A) = P(B).
Important note: Be sure to check to see that both conditions of the general multiplication rule are met. Remember that when two events are independent, knowing that one event occurred won’t help you predict if the other event will occur.
You can check to see if two events A and B are independent if P(A|B) = P(A|B^C) as well.
Multiplication rule for independent events
If events A and B are independent, then the probability that A and B both occur is P(A&B) = P(A) * P(B)
You must check to make sure events A and B are independent before using the above rule. Likewise, the addition rule P(A or B) = P(A) + P(B) holds if A and B are mutually exclusive but not otherwise. Be sure to only use them when their conditions are met.
Two mutually exclusive events can never be independent, because if one event happens, the other event isn’t guaranteed to happen.