2.7 Conditional probability
Roll a die 🎲
What is the probability the number is less than 4?
Without seeing it, we were told it landed on an odd number.
With this new information, how should we adjust our belief that the number is less than 4?
For any event \(A\), we have our initial belief of how likely it would occur. \(\rightarrow\text{P}(A)\)
Now, we learned that another event \(B\) has occurred.
With this new information, how should we adjust our belief on how likely event \(A\) would occur?
\[\text{Conditional probability:}\;\;\text{P}(A \mid B)\]
Conditional probability
For any two events \(A\) and \(B\), where \(\text{P}(B)>0\),
the conditional probability \(\text{P}(A \mid B)\) is defined as
\[\large{\text{P}(A \mid B)=\frac{\text{P}(A \cap B)}{\text{P}(B)}}\]
Exercise
Toss two coins 🪙 🪙
Without seeing it, we were told that one landed on heads.
What is the probability the other landed on tails?
Exercise
Roll two dice 🎲 🎲
- What is the probability that doubles are rolled?
- We were told that the sum is 4 or less.
- Now what is the probability that doubles are rolled?
Exercise - Simplify \(\text{P}(A \mid B)\)
- \(A\) and \(B\) are disjoint
