What is Conditional Probability?
Conditional probability is a probability of an event where another event has already occurred and is represented as P(A|B) i.e. Probability of event A given event B has already occurred. It can be calculated by multiplying P (A and B) i.e. Joint Probability of event A and event B divided by P(B), Probability of event B
Conditional probability is used only when there are two or more than two events are happening. And if there are too many events probability is calculated for each and every possible combination.
Below are the methodology followed to derive the conditional probability of event A where Event B has already occurred.
Step 1: Firstly, determine the total number of the event which makes the probability equals to 100 per cent.
Step 2: Determine the probability of event B which has already occurred by applying the probability formula i.e. P(B)= Total chances of event B happening/ All possible chances
Step 3: Next, Determine the joint probability of event A and B, P(A and B) which means chances that A and B can happen together / all possible chances of event B.
Step 4: Divide the outcome of step 3 by the outcome of step 2 to arrive at the conditional probability of event A where event B has already occurred.
Few more things to take into considerations are as below
Identify the type of events to determine the probability:-
- With Replacement: both the events are not dependent on each other which means happening of one event will not impact the probability of other events.
- Without Replacement: the events are dependent on each other, the outcome of one event will decide the outcome of other events.
- Independent Events: Probability of the second event is not influenced by the outcome of the first event, which is considered as independent events. Here conditional probability for Probability of Event A given Event B will be equal to the probability of A, i.e. P(A/B)= P(A)
- Mutually Exclusive Events: two events that can not happen together are considered as mutually exclusive events, the events which occur simultaneously. Therefore the conditional probability of one event will always be zero if other even have already happened, i.e P(A|B) = 0
Examples of Conditional Probability Formula (with Excel Template)
Let us take an example of a bag in which there are a total of 12 balls, details of balls are as below:-
- Total 5 balls are green, out of which 3 are tennis balls and 2 are footballs.
- Total 7 balls are red, out of which 2 are tennis balls and 5 are footballs.
A person X has taken out 1 ball out of the bag which turns out to be green, what is the probability of being its football.
Event 1 = whether it is a green ball or red ball
Event 2= whether it is football or tennis ball
In this case event, one has already occurred, now we have to calculate the conditional probability of event 2.
- Total Number of balls= 12
- Total number of footballs= 7
- Total number of green football= 5
P(A|B)= Probability of ball being green football
P (A and B) = Joint probability that the ball is green and it is football= Total number of green football/ Total number of balls = 2/12
P(B) = Probability of ball being green= Total green balls/ Total number of balls=5/12
Calculation of Conditional Probability
- P(A/B) = (2/12)/(5/12)
Conditional Probability will be –
- P(A|B) = (2/5)
Given are probabilities:-
- Probability of rains up to 5mm- 30%
- Probability of rains between 5mm to 15mm – 45%
- Probability of rains above 15mm- 25%
Given are the details:-
- If it rains to 5mm, out of 30%, 24% chances are there of crop production to be ruined and 6% to be better.
- If it rains between 5mm-15mm, 31.5% chances are there of crop production to be better and 13.5 % to be ruined.
- It rains above 15 mm, All the crop will be ruined.
Here we need to find the probability of crop production being better if rains are happening between 5mm- 15mm.
- Probability of rains happening between 5mm-15mm= 45%
- The joint probability of rains between 5mm-15mm and crop being better is 31.5%
Probability of rains happening between 5mm-15mm and crop production being better is as follows,
- =31.5% / 45%
- = 70%
Below are the details of the economy where the interest rate will be up or down and economic slowdown and revival are interdependent.
Figure out what is the probability that there is economic revival and interest rate will be up.
- Probability of interest rate being up = 0.61
- Probability of economic revival = .55
- Joint Probability of interest rate being up with revival economy = 0.29
Calculation of Conditional Probability
- = 0.29/0.55
- = 52.7%
If the economy has already revived and we want to predict the probability of interest rate being up = 52.7%
Relevance and Use
Conditional probability is used for risk management by assessing the probability of risk. Risk is assessed by using the probability of event and loss gave the impact has happened. It can be in several forms like assessing the financial loss of the insurance company given an event has already happened or assessing the risk of a farmer depending on weather conditions. By assessing the risk, a company/ individual can manage the risk by analyzing its impact.
Management decisions are based on future probability. Financial and other non-financial decision making that is based on what will happen in the future. Prediction of the future is just an estimate, certainty of anything is not sure. Historical data or experience is used to assess future probability.
If the impact of any one event is dependent on the other event, the conditional probability of each event is calculated with all the possible combinations.
This has been a guide to Conditional Probability and its definition. Here we discuss the formula of conditional probability calculation along with practical examples and downloadable excel template. You may also have a look at the following articles –