Conditional Probability Formula

In this section, we will discuss one of the most fundamental concepts in probability theory. It is a conditional probability. Here one question is as we obtain additional information, then how should we update probabilities of events? In this topic, we will see the methods to find the probability of one event if some other event has already occurred. This is the conditional probability formula. Let us learn it!

Conditional Probability Formula

What Is Conditional Probability?

Suppose that I pick a random day, and I also know that it is cloudy on that chosen day. In other words, what will be the probability that it rains given that it is cloudy? If C is the event that it is cloudy, then we write this probability as:\(P(R|C)\)

Which is the conditional probability of R given that C has already occurred? Therefore, the conditional probability formula will give the measure of the probability of an event given that another event has occurred. The events are commonly written as P(A|B), or sometimes P B(A).

Let us suppose that in a certain city, 23 percent of the days are rainy. Remember that it is often useful to think of probability as percentages. Then if we pick a random day, then the probability that it rains that day will be 23 percent. Thus:

\(P(R)=0.23, \textrm{where } R \textrm{ is the event that it rains on the randomly chosen day.}\)

The Formula for Conditional Probability

If A and B are two events in a given sample space S, then the conditional probability of A given B is defined as:

\(P(A|B)=\frac{P(A \cap B)}{P(B)}, \textrm{ when } P(B)>0.\)

Chain rule for conditional probability: A general statement of the chain rule for n events A_1, A_2, A_3….. is as follows:

\(P(A_1 \cap A_2 \cap \cdots \cap A_n)=P(A_1)P(A_2|A_1)P(A_3|A_2,A_1) \cdots P(A_n|A_{n-1}A_{n-2} \cdots A_1)\)

Some More Formulas

For three events, A, B, and C, with P(C)>0, we have:

1. \(P(A^c|C)=1-P(A|C) \\\)
2. \(P(\emptyset|C)=0 \\\)
3. \(P(A|C) \leq 1 \\\)
4. \(P(A-B|C)=P(A|C)-P(A \cap B|C) \\\)
5. \(P(A \cup B|C)=P(A|C)+P(B|C)-P(A \cap B|C) \\\)
6. if \(A \subset B then P(A|C) \leq P(B|C) \\\)

Solved Examples Conditional Probability Formula

Q.1: I roll a fair die. Let event A is an odd number, i.e., A={1,3,5}. Also let event B is the event with the outcome as less than or equal to 3, i.e., B= {1,2,3}. Compute P(A). What will be the probability of A given B,  P(A|B)?

Solution: This is a finite sample space, so

\(P(A)=|A||S|=|{1,3,5}|6=12\)

Now, we will find the conditional probability of A given that B occurred. If we already know B has occurred, then the outcome must be among {1,2,3}. For A to also happen the outcome must be in A∩B={1,3}. Since all rolls of a die are equally likely, then P(A|B) must be equal to

\(P(A|B)=|A∩B||B|=23\)

Q.2: The probability that the day is Friday and that a student is absent is 0.03. Since there are 5 working days in a week, the probability that it is Friday will be 0.2. Therefore, compute the probability that a student is absent on known Friday?

Solution: The formula of Conditional probability is given as:

Probability for Absent = \(P(A) = P(A\cap B) = 0.03\)

Probability for Friday = P(B) = 0.2

\(P(A|B)=\frac{P(A \cap B)}{P(B)}, \textrm{ when } P(B)>0\)

\(P(A|B)=\frac{P(A \cap B)}{P(B)}, \textrm{ when } P(B)>0\)

\(= \frac {0.03}{0.2}\) = 0.15 = 15 %