5.3. Conditional probability
Conditional probability deal with finding the probability of an event when you know that the
outcome was in some particular part of a sample space
- It is most commonly used to find a probability about a category for one variable, when we
know the outcome on another variable
- For events A and B, the conditional probability of event A, given that event B has occurred,
P( A∧B)
is: P ( A|B )=
P (B)
- P(A|B) is read as “the probability of event A, given event B”. The vertical slash represents
the word “given”. Of the times that B occurs, P(A|B) is the proportion of times that A also
occurs.
-
Multiplication rule for finding P(A and B)
- When A and B are independent events P ( A∧B )=P (A )× P (B). The definition of
conditional probability provides a more general formula for P(A and B) that holds regardless
of whether A and B are independent ↓
- For events A and B, the probability that A and B both occur, equals
P ( A∧B )=P (B)× P( A∨B)
- Applying the conditional probability formula to P(B|A), we also see that
P ( A∧B )=P ( A )× P (B∨ A)
Sampling without replacement = in a sampling process, once a subject is selected from a
population, they are not eligible to be selected again.
- Probabilities of potential outcomes depend on the previous outcomes.
- Conditional probabilities are then used in finding probabilities of the possible samples.
Independent events defined using conditional probability
- Events A and B are independent if the probability that one occurs is not affected by whether
the other event occurs.
- Events A and B are independent if P ( A|B )=P ( A ) , or equivalently, if P ( B| A )=P( B). If
either holds, then the other does too.
Conditional probability deal with finding the probability of an event when you know that the
outcome was in some particular part of a sample space
- It is most commonly used to find a probability about a category for one variable, when we
know the outcome on another variable
- For events A and B, the conditional probability of event A, given that event B has occurred,
P( A∧B)
is: P ( A|B )=
P (B)
- P(A|B) is read as “the probability of event A, given event B”. The vertical slash represents
the word “given”. Of the times that B occurs, P(A|B) is the proportion of times that A also
occurs.
-
Multiplication rule for finding P(A and B)
- When A and B are independent events P ( A∧B )=P (A )× P (B). The definition of
conditional probability provides a more general formula for P(A and B) that holds regardless
of whether A and B are independent ↓
- For events A and B, the probability that A and B both occur, equals
P ( A∧B )=P (B)× P( A∨B)
- Applying the conditional probability formula to P(B|A), we also see that
P ( A∧B )=P ( A )× P (B∨ A)
Sampling without replacement = in a sampling process, once a subject is selected from a
population, they are not eligible to be selected again.
- Probabilities of potential outcomes depend on the previous outcomes.
- Conditional probabilities are then used in finding probabilities of the possible samples.
Independent events defined using conditional probability
- Events A and B are independent if the probability that one occurs is not affected by whether
the other event occurs.
- Events A and B are independent if P ( A|B )=P ( A ) , or equivalently, if P ( B| A )=P( B). If
either holds, then the other does too.
