Example:Weighted coin tree

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Consider the experiment where we choose coin a 1/3 of the time, and coin b 2/3 of the time, and where coin a lands heads 3/4 of the time and coin b lands heads 1/2 of the time.

We can draw a tree to organize these events into a tree:


Probability-tree.svg


The vertices in the tree represent events; the event [math]E_1 [/math] is a child of [math]E_2 [/math] if [math]E_1 \href{/cs2800/wiki/index.php/%E2%8A%86}{⊆} E_2 [/math]. The number on the edge from [math]E_1 [/math] to [math]E_2 [/math] is the conditional probability of [math]E_2 [/math] given [math]E_1 [/math].

The probability of an event in the tree can be found by multiplying the probabilities on the path leading to that event. This comes from the definition of conditional probability: if [math]E_1 \href{/cs2800/wiki/index.php/%E2%8A%86}{⊆} E_2 [/math] then [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(E_1) = \href{/cs2800/wiki/index.php/Pr}{Pr}(E_1 \href{/cs2800/wiki/index.php/%E2%88%A9}{∩} E_2) = \href{/cs2800/wiki/index.php/Pr}{Pr}(E_1 \href{/cs2800/wiki/index.php/%5Cmid}{\mid} E_2)\href{/cs2800/wiki/index.php/Pr}{Pr}(E_2) [/math].