The probability of an Event ''A'' conditional on another event ''B'' is generally different from the probability of ''B'' conditional on ''A''. However, there is a definite relationship between the two, and Bayes' theorem is the statement of that relationship.
As a formal and Frequentist Probability discuss these debates at greater length.
Bayes' theorem relates the conditional and marginal probabilities of Stochastic Events ''A'' and ''B'':
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"http://wwwinformationdelightinfo/encyclopedia/entry/likelihood" class="copylinks">Likelihood of ''A'' given ''B'' for a fixed value of ''B''
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"http://wwwinformationdelightinfo/encyclopedia/entry/conditional_probability" class="copylinks">Conditional Probability '' of ''A'', given ''B'' It is also called the Posterior Probability because it is derived from or depends upon the specified value of ''B''
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In words: the posterior probability is proportional to the prior probability times the likelihood.
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rac{\Pr(A \cap B)}{\Pr(B)}</math>
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rac{\Pr(A \cap B)}{\Pr(A)} \!</math>
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\Pr(A \cap B) = \Pr(BA)\, \Pr(A) \!</math>
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rac{\Pr(BA)\,\Pr(A)}{\Pr(B)} \!</math>
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\Pr(A, B) + \Pr(A^C, B) = \Pr(BA) \Pr(A) + \Pr(BA^C) \Pr(A^C)\,</math>
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rac{\Pr(B A)\, \Pr(A)}{\Pr(BA)\Pr(A) + \Pr(BA^C)\Pr(A^C)} , \!</math>
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rac{\Pr(B A_i)\, \Pr(A_i)}{\sum_j \Pr(BA_j)\,\Pr(A_j)} , \!</math>
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O(A) \cdot \Lambda (AB) </math>
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rac{\Pr(AB)}{\Pr(A^CB)} \!</math>
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rac{L(AB)}{L(A^CB)} = rac{\Pr(BA)}{\Pr(BA^C)} \!</math>
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rac{f(x,y)}{f(y)} = rac{f(yx)\,f(x)}{f(y)} \!</math>
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rac{f(yx)\,f(x)}{\int_{-\infty}^{\infty} f(yx)\,f(x)\,dx}
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''y'',
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 ''L''(''x''''y'') is (as a function of ''x'') the likelihood function of ''X'' given ''Y''=''y'',
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rac{\Pr(A) \, \Pr(BA) \, \Pr(CA,B)}{\Pr(B) \, \Pr(CB)} \!</math>
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rac{\Pr(A,B,C)}{\Pr(B,C)} = rac{\Pr(A,B,C)}{\Pr(B) \, \Pr(CB)} = </math>
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rac{\Pr(CA,B) \, \Pr(A,B)}{\Pr(B) \, \Pr(CB)} = rac{\Pr(A) \, \Pr(BA) \, \Pr(CA,B)}{\Pr(B) \, \Pr(CB)} </math>
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P(TD)\,P(D) + P(TD^C)\,P(D^C)
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rac{P(TD)\,P(D)}{P(TD)\,P(D) + P(TD^C)\,P(D^C)}
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rac{099 imes 0001}{099 imes 0001 + 005 imes 0999}
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rac{\Pr(B A) \Pr(A)}{\Pr(B)} = rac{075 imes 05}{0625} = 06</math>
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"1" cellspacing="0" cellpadding="8"
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10, m=7) =
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7 r, n=10) \, f(r)} {\int_0^1 f(m=7r, n=10) \, f(r) \, dr} \!</math>
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 ''f''(''m'' = 7''r'', ''n'' = 10), we can compute the posterior probability density function ''f''(''r''''n'' = 10, ''m'' = 7)
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 ''P''(''m'' = 7''r'', ''n'' = 10,) for such a problem is just the probability of 7 successes in 10 trials for a Binomial Distribution
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7 r, n=10) = {10 \choose 7} \, r^7 \, (1-r)^3 </math>
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7r, n=10) \, f(r) \, dr = \int_0^1 {10 \choose 7} \, r^7 \, (1-r)^3 \, 1 \, dr = {10 \choose 7} \, rac{1}{1320} \!</math>
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10, m=7) =
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