Sample Space for simple Probability Experiment

Question

Imagine three empty boxes. You throw a "fair" coin. If the coin display head, you put a ball inside one of the three boxes whereas the probability for box one is $p\in(0,1)$ and for box two and three it is $\frac{1-p}{2}$.

With which probability (in dependence of p) do we have a ball in the first box?

Now, intuitively one would say $\frac{p}{2}$, which is of course correct, but I'm more interested in actually writing it down in a pedantic way.

So let $H:=\text{"Coin display head"}$ and $B_i:=\text{"Ball is in box i"}$ with $i=1,2,3$.

We know $P[H]=P[H^C]=1/2$ since we have a "fair" coin. Furthermore, we have apparently:

$P[B_1|H]=p, \quad P[B_i|H]=\frac{1-p}{2}, \ i=2,3$ and $P[B_i|H^C]=0, \ i=1,2,3$

Now, I basically fail to see how to properly calculate e.g. $P[B_1|H]=\frac{P[B_1 \cap H]}{P[H]}$ and the reason for that is, that I'm confused on how to calculate $P[B_1 \cap H]$ and the reason for that is, that we never really modeled the sample space for the experiment.

Of couse, it makes intuitively sense. To be able to have the event $B_1$ we need to have thrown a coin which displays head, i.e. $B_1$ implies $H$. But I'd like to actuall write down these sets.

How would one do that? I'm confused on how to model the sample space $\Omega$ if we have events that kind of imply some events?

My take would be: $\Omega:=\{ \{B_1, H\}, \{B_2, H\}, \{B_3, H\}, H, H^C\}$ (whereas I really really dislike the notation of $H^C$ for "Coin displays number." since it it just wrong. We get what is meant, but with the given $\Omega$ above, it'd be wrong... but anyway, as you can see, my $\Omega$ also just seems to be absolute trash.

Can someone give me an example of a properly modeled sample space? I don't want to work with "intuition" since I really think probability is heavy anti-intuition.

Answer 1

The sample space $\Omega$ is $$\Omega = \{B_1, B_2, B_3, T\}$$ where $T$ (tails) stands for your $H^C$.

The elements $\omega_i$ of the sample space $\Omega$ must be

• mutually exclusive (if $\omega_i$ occurred, then $\omega_j, j\neq i$ can't occur)
• collectively exhaustive (one element $\omega_i$ always occurs in the probabilistic experiment)

The sample space $\Omega = \{B_1, B_2, B_3, T\}$ satisfies these requirements, therefore is a correct sample space for your experiment.