# Category Archives: Bounding Probabilies

## Pathway enrichment

A list of $$100$$ genes are known to be part of the oxidative phosphorylation pathway.

My friend a molecular biologist screened the activity of $$5000$$ genes in both diabetics and normal individuals.

He/she found $$500$$ genes that were more active in normal individuals than diabetics.

Of these genes $$60$$ of them belong to the list of genes that are part of the oxidative phosphorylation pathway.

What is the probability of this even happening randomly ? What is the scientific question behind the probability problem ?

## Inclusion of origin

Draw $$n$$ points from the uniform distribution on the circle and draw the convex hull around these points. What is the probability that the origin (center of the circle) is contained in the convex hull ?

[From: The Probabilistic Method by Alon and Spencer]

## Estimating differences of independent draws

1. Show that if $$X$$ and $$Y$$ are independent random variables, then
$\mathrm{Var}(X- Y)=\mathrm{Var}(X+Y)$
2. Let $$D_1$$ and $$D_2$$ represents two draws at random with replacement from a population, with $$\mathbf{E}D_1=10$$ and $$\mathbf{SD}(D_1)_1=2$$. Find a number $$c$$ so that
$\mathbf{P}(|D_1 -D_2| < c) \geq .99$

From [Pittman p. 203, #15]