Conditional probability – Probability of one event given another has occurred.
The conditional probability of A given B is denoted by \(P(A | B)\)
How do we find the conditional probabilty, \(P(A | B)\)
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
In general, \(P(A | B) \neq P(B | A)\).
Consider the probability of a flight departing on time \(P(D)\) is 0.83, and the probability of a flight arriving on time \(P(A)\) is 0.82. The probability that a flight departs and arrives on time \(P(D \cap A)\) is 0.78.
Find the probability
The first question asks for \(P(A | D)\), which we can find using \(\frac{P(A \cap D)}{P(D)} = \frac{0.78}{0.83} = 0.94\).
The second question asks us it in reverse: \(P(D | A) = \frac{0.78}{0.82} = 0.95\).
Notice the two are not necessarily the same.
There are 20 balls in a box, either red or green, either rubber or plastic. If we know that 12 balls are red and 9 balls are rubber and 4 balls are green and rubber:
First question:
# of plastic balls = 20 - 9 = 11
# of plastic and green balls = (20 - 12) - 4 = 4
# of plastic and red balls = 11 - 4 = 7
Second question:
\(P(P | R) = \frac{P(P \cap R)}{P(R)}\)
\(\frac{7}{12}\)
Third question:
\(P(P | G) = \frac{P(P \cap G)}{P(G)}\)
\(\frac{4}{20 - 12}\)
\(\frac{1}{2}\)
When two events A and B are independent:
\(P(A \cap B) = P(A) * P(B)\)
Also complements like A’ and B’ are independent of each other:
Insurance companies assume that there is a difference between gender and your likelihood of getting into an accident which is why women generally have lower insurance rates than men. We did a study to see the number of accidents that have occurred according to gender. We found:
Does this study indicate that the likelihood of one to get into an accident depends on gender?
Suppose we have two events, A and B:
The question is, are these two events indepdendent?
\(P(A) = 1 - 0.35 = 0.65\) \(P(B) = 0.6\) \(P(A \cap B) = 0.39\). This can be determined by \(P(A) + P(B) - P(A \cup B)\).
Notice that \(P(A \cap B) = P(A) * P(B)\). This means that events A and B are independent. In other words, the event a person gets into an accident does not depend on the gender of the person.
\(P(A \cap B) = P(A) * P(B|A)\)