Conditional Probability Calculator
This Conditional Probability Calculator computes the probability of Event A happening, given that Event B has already happened (i.e. P(A|B)). It uses the joint probability of A and B and the probability of B to find the conditional probability. Alternatively, the calculator can help you find the the probability of Event B happening, given that Event A has occurred, denoted as P(B|A).
The key feature of this calculator is that it goes beyond providing you with an accurate answer by providing you with a clear step-by-step workings. This ensures you not only get accurate results but also understand how to solve conditional probability problems.
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How to Use the Conditional Probability Calculator
Finding conditional probability is simple with this tool. You don’t need to memorize formulas because the calculator does all the work for you. You only need to follow these 4 simple steps and the calculator does the heavy-lifting for you:
- Select the probability you want to find (select either P(A|B) or P(B|A) )
- Enter the required values:
- The joint probability of both events, P(A and B).
- The probability of P(A) or P(B), depending on your choice.
- Click “Calculate” button
- The calculator will show you the result and a step-by-step breakdown of how the answer was found.
Note. Using the conditional probability tool is the easiest and fastest way to get accurate conditional probabilities, while understanding the concepts.
What is Conditional Probability?
Conditional probability is the chance that one event occurs after another related event has already happened.
Let’s assume we have two events, A and B. If the probability of A changes when B occurs, then A and B are called dependent events. This dependency is what conditional probability measures.
In simple terms, conditional probability tells us the chance that Event A happens given that Event B has already happened. Thus, the conditional probability formula is given by:
$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$
This means the probability of A given B equals the probability that both A and B happen, divided by the probability that B happens.
Similarly, $P(B|A) = \frac{P(A \text{ and } B)}{P(A)} $
Where:
- P(A) is the probability that Event A occurs.
- P(B) is the probability that Event B occurs.
- P(A and B) is the probability that both A and B occur together.
For example, if A is the event that “it rains” and B is the event that “the sky is cloudy,” then P(A|B) represents the chance of rain, given that the sky is cloudy. These two events are dependent, because the chance of rain changes when you know the sky is cloudy.
However, if these two events are independent, then the occurrence of one does not affect the other. In that case:
P(A∣B)=P(A) and P(B/A) = P(B)
Conditional Probability Solved Examples
Conditional probability becomes easier to understand when we use real-world examples. It helps us find how likely one event is to happen after another event has already occurred. Let’s go through an real-world examples to see how it works in practice.
Scenario: A hospital conducted a study to see the relationship between symptom A and test B.
They found that:
- 30% of patients had both symptom A and a positive test B result.
- 60% of patients tested positive for test B.
Using this information, what is the probability that a patient has symptom A given that they tested positive for test B?
Solution
Let:
- A be the event patients had symptom A
- B be the event that patients test positive for event B
From the scenario, then P(A and B) = 0.30 and P (B) = 0.60
Applying the conditional probability formula, we have:
$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$
Substituting the known probabilities in the formula, we have:
$P(A|B) = \frac{0.3}{0.6}$
=0.5
Interpretation: There is a 50% chance that a patient has symptom A, given that they tested positive for test B.
Alternatively, using our free conditional probability calculator, you get similar results with step-by-step explanations, as shown below:
Frequently Asked Questions
A Conditional Probability Calculator is an online tool that helps you find the probability of one event occurring given that another event has already happened. It calculates values like P(A|B) or P(B|A) and shows the step-by-step solution using the correct formula.
You can calculate conditional probability using the formula:
$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$.
This means you divide the probability of both events happening (P(A and B)) by the probability of the given event (P(B)).
– P(A|B) means the probability that event A occurs given that event B has already occurred.
– P(B|A) means the probability that event B occurs given that event A has already occurred.
Note that these two values are not always the same. They depend on which event is known first.
You should use a conditional probability solver when you need to find the relationship between two dependent events. It is especially useful in subjects like statistics, data analysis, probability theory, and even in real-life situations such as medical testing or risk assessment.
P(A and B) represents the joint probability. In other words, it is the likelihood that both events A and B occur together. It is used in the conditional probability formula to calculate either P(A|B) or P(B|A).
The step-by-step dependent probability calculator not only gives the final answer but also shows how it was calculated. It explains each step clearly, making it easier for students, teachers, and professionals to understand how to find conditional probability using real examples and simple formulas.