11.1.4 - Conditional Probabilities and Independence

In 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 Section

If a card is randomly drawn from a standard 52-card 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 Section

Data concerning two categorical variables can be displayed in a contingency table. 

Contingency Table of Gender and Pass
  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.