upshard

Save your progress

Everything on Upshard is free and readable without an account. Signing in saves what you have finished, your bookmarks and your notes — and nothing else.

or

We store your email, your progress, your bookmarks and your notes. Nothing else, ever. What we do with it.

Your account

Delete your account?

This removes your progress, bookmarks and notes immediately and permanently. It cannot be undone. Everything on Upshard stays free to read without an account.

Law of total probability

Probability · Formula · Not yet linked from a lesson

P(A)=∑iP(A∣Bi) P(Bi)P(A) = \sum_{i} P(A \mid B_i)\, P(B_i)

Splits a messy probability into cases you can actually compute, then weights each by how likely that case was. It is the denominator in Bayes' theorem whenever the evidence is not handed to you directly.

Symbols

BiB_i
A set of cases that are exclusive and cover everything.
P(A∣Bi)P(A \mid B_i)
The probability of A within case i.
Esc
↑↓ navigate↵ open? shortcuts

Keyboard shortcuts

Press Esc to close

Anywhere

⌘K
Search and commands
/
Search
?
This sheet
Esc
Close

Reading a lesson

J
Next lesson
K
Previous lesson
B
Bookmark