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Pondering a question on conditional probability.

Posted by peeterjoot on December 27, 2011

[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

A bit of confusion.

In [1] section 1.5 while discussing statistical uncertainty is a mention of conditional probability. Once told that a die only rolls numbers up to four, we have a conditional probability for the die of

\begin{aligned}P(i | i \le 4) = \frac{P[i U(i \le 4)]}{P(i \le 4)}= \frac{\frac{1}{{6}}}{\frac{4}{6}} = \frac{1}{{4}}.\end{aligned} \hspace{\stretch{1}}(1.1)

I was having trouble understanding the numerator in this expression. An initial confusion was, “what is this U”. I came to the conclusion that this is just a typo, and was meant to be set intersection, as in

\begin{aligned}P(i | i \le 4) = \frac{P[i \cap (i \le 4)]}{P(i \le 4)}= \frac{\frac{1}{{6}}}{\frac{4}{6}} = \frac{1}{{4}}.\end{aligned} \hspace{\stretch{1}}(1.2)

The denominator makes sense. I picture a sample space with 6 points as in figure (\ref{fig:conditionalProbStatMechalProbStatMechFig1})

Sample space for a die.

The points 1,2,3,4 represent the $i \le 4$ subspace, as in figure (\ref{fig:conditionalProbStatMechalProbStatMechFig2})

So we have a $4/6$ probability for that compound event.

I found it easy to get mixed up considering the numerator. I was envisioning, as in figure (\ref{fig:conditionalProbStatMech:conditionalProbStatMechFig3})

a set of six intersecting with the set of points $1,2,3,4$, which is just that set of four. To clear up the confusion, imagine instead that we are asking about the probability of finding the $i = 2$ face in the dice roll, given the fact that the die only rolls $i = 1, 2, 3, 4$. Our intersection sample space for that event is then shown in figure

The intersection is just the single point, so we have a $1/6$ probability for that compound event. The final result makes sense, since if we are looking for the probability for any of the $i = 1, 2, 3, 4$ events given that the die will only roll one of these values, we should have a value of $1/4$ as found through the compound probability formula.

References

[1] E.A. Jackson. Equilibrium statistical mechanics. Dover Pubns, 2000.