EX 2.11 toss a thumbtack

Consider tossing a thumbtack in the air. When it comes to rest on the ground, either its point will be up (the outcome $U$) or down (the outcome $D$). The sample space for this event is therefore $\mathcal{S}=\{U,D\}$. The axioms specify $P\left(\mathcal{S}\right)=1$, so the probability assignment will be completed by determining $P\left(U\right)$ and $P\left(D\right)$ . Since $U$ and $D$ are disjoint and their union is $\mathcal{S}$, the foregoing proposition implies that

$$1=P\left(\mathcal{S}\right)=P\left(U\right)+P\left(D\right)$$

It follows that $P\left(D\right)=1-P\left(U\right)$.

• One possible assignment of probabilities is $P\left(U\right)=.5$, $P\left(D\right)=.5$
• another possible assignment is $P\left(U\right)=.75$, $P\left(D\right)=.25$.

In fact, letting $p$ represent any fixed number between 0 and 1, $P\left(U\right)=p$ , $P\left(D\right)=1-p$ is an assignment consistent with the axioms.