Bayes’ Theorem is a fundamental concept in probability theory and statistics. Bayes’ Theorem is a mathematical formula that helps us update our probabilities when new evidence becomes available.
How to use Bayes Theorem Calculator with steps by steps Procedure?
- Enter the Probability of an Event A1 ($P(A_k)$)
- Enter the Probability of B under the condition of A1 ($P(B|A_k)$)
- Enter the Probability of an Event A2 ($P(A_k)$)
- Enter the Probability of B under the condition of A2 ($P(B|A_k)$)
- Click on “Calculate” button to calculate Bayes probability
- Calculate Total Probability ($P(B)$)
- Calculate Probability of an Event A1 given B ($P(A_1|B)$)
- Calculate Probability of an Event A2 given B ($P(A_2|B)$)
Bayes Theorem Calculator
| Bayes Probability Calculator | |||
|---|---|---|---|
| Event | $P(A_k)$ | $P(B|A_k)$ | |
| Event $A_1$ | |||
| Event $A_2$ | |||
| Probability $P(B)$ | |||
| Results | |||
| $P(A_1|B)$ | |||
| $P(A_2|B)$ | |||
What is Bayes Rule?
Bayes’ theorem allows us to calculate the updated probability of an event happening, given new evidence.
The formula for Bayes’ Theorem is as follows:
$$ \begin{aligned} P(A|B) &=\frac{P(A)P(B|A)}{P(B)}\\ &= \frac{P(A)P(B|A)}{P(A)P(B|A)+P(A^\prime)P(B|A^\prime)} \end{aligned} $$
where
- P(A) is the prior probability of hypothesis A being true, which is based on initial knowledge or beliefs.
- P(B) is the probability of observing evidence B, regardless of the truth or falsity of hypothesis A.
- P(A∣B) represents the probability of hypothesis A being true given the evidence B.
- P(B|A) is the probability of observing evidence B if hypothesis A is true.
For more than one event:
Let $A_1, A_2, \cdots, A_n$ are mutually exclusive and exhaustive
events of the sample space $S$ and if $P(A_i)\neq 0$, $i=1,2,\cdots,n$, then for any event $B$ of the sample space $S$
$$ \begin{equation*} P(A_i/B) =\frac{P(A_i) A(B/A_i)}{\sum_{i=1}^n P(A_i) A(B/A_i)} \end{equation*} $$
for $i=1,2,\cdots , n$.
