Probability of a Probability

Calculus Level 5

Consider a random event $E$ with probability $p:=\mathbb{P}[E]$ . Consider a condition $C$ , which is also a random variable taking values in some space $\Omega$ , and upon which $E$ depends. That is, $\mathbb{P}[E\mid C=c]$ may different from $\mathbb{P}[E]$ for different values $c$ .

Assume that the current a priori probability of the Event $E$ is extremely small. We know further nothing about the distribution of $C$ , but we are looking for favourable conditions $c$ such that the probability of $E$ being sufficiently larger than the a priori probability, namely at least $10\cdot p$ .

What can we say about the probability of the probability of $E$ being increased in this way? That is, investigate $\mathbb{P}_{C}[\{c\in\Omega:\mathbb{P}[E\mid C=c]\geq 10p\}]$ .

This probability is undefined, as the nested event is non-measurable. It is at most

1/10

. It is at least

1/10

. Impossible to tell without knowledge of the distribution of

C

. It depends on the Event itself. It is exactly

1/10

2 solutions

R Mathe
May 27, 2018

Answer. At most $\frac{1}{10}$ .

Solution. I shall compute more generally. Let

$E_{q}:=\{c\in\Omega:\mathbb{P}[E\vert C=c]\geq q\}.$

The event is measurable, but I won’t go into the formalities. One computes using condition probabilities (this is well defined in Measure theory under very natural circumstances):

$\begin{array}{c}[t]{rcl} p=\mathbb{P}[E] &= &\int_{c\in\Omega}\mathbb{P}[E\vert C=c]~\text{d}\mathbb{P}_{C}(c)\\ &\geq &\int_{c\in E_{q}}\mathbb{P}[E\vert C=c]~\text{d}\mathbb{P}_{C}(c)\\ &\geq &\int_{c\in E_{q}}q~\text{d}\mathbb{P}_{C}(c) \quad\text{since }\mathbb{P}[E\vert C=c]\geq q\text{ on }E_{q}\\ &= &q\cdot\mathbb{P}_{C}[E_{q}].\\ \end{array}$

Hence $\mathbb{P}_{C}[E_{q}]\leq\frac{p}{q}$ . Thus in general we have

$\mathbb{P}_{C}[\{c\in\Omega:\mathbb{P}[E\vert C=c]\geq q\}]\leq\frac{p}{q}$ .

For our particular problem, $q=10\cdot p$ , hence the sought out probability is at most $\frac{p}{10p}=\frac{1}{10}$ .

Of course, if $p=0$ (which the problem has not excluded - we are just told that $p$ is extremely small). $\mathbb{P}[E_{pq}] = 1$ for all $q > 0$ .

Mark Hennings - 2 years, 6 months ago

Abhishek Sinha
May 28, 2018

This is a simple application of Markov Inequality , once we realize that $\mathbb{E}_C(\mathbb{P}(E|C))=p.$

Probability of a Probability

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