Experiment/Examples
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Examples of Experiments
Throwing a $6$-Sided Die
Let $\EE$ be the experiment of throwing a standard $6$-sided die.
- The sample space of $\EE$ is $\Omega = \set {1, 2, 3, 4, 5, 6}$.
- Various events can be identified:
- $(1): \quad$ The result is $3$:
- The event space of $\EE$ is: $\Sigma = \set 3$.
- $(2): \quad$ The result is at least $4$:
- The event space of $\EE$ is: $\Sigma = \set {\forall \omega \in \Omega: \omega > 4}$.
- $(3): \quad$ The result is a prime number:
- The event space of $\EE$ is: $\Sigma = \set {2, 3, 5}$.
- The probability measure is defined as:
- $\forall \omega \in \Omega: \map \Pr \omega = \dfrac 1 6$
Tossing $2$ Coins
Let $\EE$ be the experiment of tossing $2$ coins.
The sample space of $\EE$ is:
- $\Omega = \set {\tuple {\mathrm H, \mathrm H}, \tuple {\mathrm H, \T}, \tuple {\T, \mathrm H}, \tuple {\T, \T} }$
where $\mathrm H$ denotes heads and $\T$ denotes tails.
Suppose we are interested only in whether the coins fall alike ($\mathrm A$) or different ($\mathrm D$).
Then the sample space of $\EE$ is:
- $\Omega = \set {\mathrm A, \mathrm D}$