11.1.4  Conditional Probabilities and Independence
11.1.4  Conditional Probabilities and IndependenceIn Lesson 2 you were introduced to conditional probabilities and independent events. These definitions are reviewed below along with some examples.
Recall that if events A and B are independent then \(P(A) = P(A \mid B)\). In other words, whether or not event B occurs does not change the probability of event A occurring.
 Conditional Probability

The probability of one event occurring given that it is known that a second event has occurred. This is communicated using the symbol \(\mid\) which is read as "given."
For example, \(P(A\mid B)\) is read as "Probability of A given B."
 Independent Events
 Unrelated events. The outcome of one event does not impact the outcome of the other event.
Example: Queens & Hearts
If a card is randomly drawn from a standard 52card deck, the probability of the card being a queen is independent from the probability of the card being a heart. If I tell you that a randomly selected card is a queen, that does not change the likelihood of it being a heart, diamond, club, or spade.
Using a conditional probability to prove this:
\(P(Queen) = \dfrac{4}{52}=0.077\)
\(P(Queen \mid Heart) = \dfrac {1}{13} = 0.077\)
Example: Gender and Pass Rate
Data concerning two categorical variables can be displayed in a contingency table.
Pass  Did Not Pass  Total  
Men  6  9  15 
Women  10  15  25 
Total  16  24  40 
If gender and passing are independent, then the probability of passing will not change if a case's gender is known. This could be written as \(P(Pass) = P(Pass \mid Man)\).
\(P(Pass) = \dfrac{16}{40} = 0.4\)
\(P(Pass \mid Man) = \dfrac{6}{15}=0.4\)
In this sample, gender and passing are independent.