Bayes' Theorem/Examples/Arbitrary Example 3

Example of Use of Bayes' Theorem

Let $C_1$ be a a double-headed coin.

Let $C_2$ be a fair coin.

Let $A_1$ denote the event of tossing $C_1$.

Let $A_2$ denote the event of tossing $C_2$.

Suppose one of $C_1$ and $C_2$ is chosen at random with $\map \Pr {A_1} = \dfrac 1 2$ and $\map \Pr {A_2} = \dfrac 1 2$.

Let $B$ be the event of landing heads.

What is the probability that it was $C_1$ that was tossed?

We have:

$\ds \condprob B {A_1}$

$=$

$\ds 1$

$\ds \condprob B {A_2}$

$=$

$\ds \dfrac 1 2$

where $\condprob B A$ denotes the conditional probability of $B$ given $A$.

Thus:

$\ds \condprob B {A_1}$

$=$

$\ds \dfrac {\condprob B {A_1} \map \Pr {A_1} } {\condprob B {A_1} \map \Pr {A_1} + \condprob B {A_2} \Pr {A_2} }$

$\ds $

$=$

$\ds \dfrac {1 \times \frac 1 2} {1 \times \frac 1 2 + \frac 1 2 \times \frac 1 2}$

$\ds $

$=$

$\ds \dfrac 2 3$

So the probability that $C_1$ was tossed is $\dfrac 2 3$.

$\blacksquare$

2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Bayes' Theorem
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Bayes' Theorem