How does Bayes’s theorem apply to drawing red and black balls from an urn?

Bayes’s theorem allows for the calculation of inverse probabilities, such as determining the likelihood of an event on a first draw based on the known outcome of a second draw.

In an experiment involving an urn with $r$ red balls and $b$ black balls, suppose two balls are drawn sequentially without replacement. Let $A$ be the event of drawing a red ball first, and $B$ be the event of drawing a red ball second. The probability of the first draw being red, given that the second draw is known to be red, is calculated using the theorem as follows:

• $P(A)=r/(r+b)$
• $P(B|A)=(r-1)/(r+b-1)$
• $P(B|A^c)=r/(r+b-1)$

By applying Bayes’s theorem, the probability $P(A|B)$ equals $(r-1)/(r+b-1)$. This demonstrates how conditional probability updates our knowledge of an initial state once subsequent evidence is observed.12