An Essay Towards Solving a Problem in the Doctrine of Chances

"An Essay Towards Solving a Problem in the Doctrine of Chances" is a work on the mathematical <a href="/facts/Theory_of_probability/mFBn51rE">theory of probability</a> by <a href="/facts/Thomas_Bayes/Btz9DmbC">Thomas Bayes</a>, published in 1763, two years after its author's death, and containing multiple amendments and additions due to his friend <a href="/facts/Richard_Price/cHMnPYej">Richard Price</a>. The title comes from the contemporary use of the phrase "doctrine of chances" to mean the theory of probability, which had been introduced via <a href="/facts/The_Doctrine_of_Chances/FyF2UuqI">the title of a book</a> by <a href="/facts/Abraham_de_Moivre/m9hpa6Mb">Abraham de Moivre</a>. Contemporary reprints of the essay carry a more specific and significant title: A Method of Calculating the Exact Probability of All Conclusions Founded on Induction.
The essay includes theorems of <a href="/facts/Conditional_probability/QcN2UERV">conditional probability</a> which form the basis of what is now called <a href="/facts/Bayes%2527s_Theorem/wQqFB2Dt">Bayes's Theorem</a>, together with a detailed treatment of the problem of setting a <a href="/facts/Prior_probability/JQKAD4o0">prior probability</a>.
Bayes supposed a sequence of independent experiments, each having as its outcome either success or failure, the probability of success being some number p between 0 and 1. But then he supposed p to be an uncertain quantity, whose probability of being in any interval between 0 and 1 is the length of the interval. In modern terms, p would be considered a <a href="/facts/Random_variable/TwTBXnLT">random variable</a> <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">uniformly distributed</a> between 0 and 1. <a href="/facts/Conditional_distribution/0eGm3P9W">Conditionally on the value of p</a>, the trials resulting in success or failure are independent, but unconditionally (or "<a href="/facts/Marginal_distribution/U9XBWAd1">marginally</a>") they are not. That is because if a large number of successes are observed, then p is more likely to be large, so that success on the next trial is more probable. The question Bayes addressed was: what is the conditional probability distribution of p, given the numbers of successes and failures so far observed. The answer is that its <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> is

f
        (
        p
        )
        =
        
          
            
              (
              n
              +
              1
              )
              !
            
            
              k
              !
              (
              n
              −
              k
              )
              !
            
          
        
        
          p
          
            k
          
        
        (
        1
        −
        p
        
          )
          
            n
            −
            k
          
        
        
           for 
        
        0
        ≤
        p
        ≤
        1
      
    
    {\displaystyle f(p)={\frac {(n+1)!}{k!(n-k)!}}p^{k}(1-p)^{n-k}{\text{ for }}0\leq p\leq 1}

(and ƒ(p) = 0 for p < 0 or p > 1) where k is the number of successes so far observed, and n is the number of trials so far observed. This is what today is called the <a href="/facts/Beta_distribution/DbJk4eeV">Beta distribution</a> with parameters k + 1 and n − k + 1.

An Essay Towards Solving a Problem in the Doctrine of Chances open-in-new

An Essay Towards Solving a Problem in the Doctrine of Chances