Amelia's exam

Amelia is about to attend an exam. Her examiner will be chosen between Prof. $X$ and Prof. $Y$. If Professor $X$ is chosen, Amelia knows she will pass with probability $0.3$ ($X$ is a well-known strict and grumpy person - Amelia is generally OK with a strict person, but she doesn't enjoy an atmosphere of grumpiness as it would stress her). However, if Professor $Y$ is chosen, Amelia knows she will have a relaxing time and pass with probability $0.8$.¹

To choose Amelia's examiner, Parinya is tossing a (biased) coin which chooses $X$ with probability $p$ and $Y$ with probability $1-p$ (for $p =0.6$). Let us estimate Amelia's chances of passing this exam and define the relevant probability spaces formally.

This is an example where we use probability to model real life. How should we think about the probability space? The probability space $\Omega$ can be written as $\Omega = \Omega_X \cup \Omega_Y$ where $\Omega_X$ and $\Omega_Y$ are disjoint events (i.e., Amelia never needs both examiners). We can interpret $\Omega_X$ as containing atomic events determining Amelia's situation (e.g., each atomic event can encode examiner's mood and Amelia's physical condition). Let $A \subseteq \Omega$ be the event that Amelia passes. Then

${\mathbb P}[A] = {\mathbb P}[A \mid \Omega_X]\cdot {\mathbb P}[\Omega_X] + {\mathbb P}[A \mid \Omega_Y]\cdot {\mathbb P}[\Omega_Y]= (0.3)(0.6)+ (0.8)(0.4) = 0.5$

In other words, this exam is just like tossing a (n unbiased) coin.

Footnotes

1. "Let me have men about me that are fat, sleek-headed men and such as sleep a-nights. Yond Cassius has a lean and hungry look, he thinks too much; such men are dangerous." - Julius Caesar, in a conversation with Mark Antony, appeared in Shakespear. ↩