Question

A sample space consists of three sample points,${\mathbf{S}}_{\mathbf{1}}\mathbf{,}{\mathbf{S}}_{\mathbf{2}}\mathbf{,}{\mathbf{S}}_{\mathbf{3}}$, with$\mathbf{P}\mathbf{(}{\mathbf{S}}_{\mathbf{1}}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}$, and$\mathbf{P}\mathbf{(}{\mathbf{S}}_{\mathbf{2}}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2}$.

1. Assign a probability to${\mathbf{S}}_{\mathbf{3}}$in a way that the three sample points define a valid sample space obeying the two probability rules.
2. If an event$\mathbf{A}\mathbf{=}\mathbf{\{}{\mathbf{S}}_{\mathbf{2}}\mathbf{,}{\mathbf{S}}_{\mathbf{3}}\mathbf{\}}$, find P(A).

Short answer

Therefore, the results are

1. $P\left(S_3\right)=0.7$
2. $P\left(A\right)=0.5$

Step-by-step solution

Given information 

Here A sample space consists of three sample points,$S_1,S_2,S_3$ , with $P\left(S_1\right)=0.1$, and$P\text{(}{S}_2\text{)}=0.2$ .

Define a valid space

a.

According to the question

Assign a probability to$S_3$ in a way that the three sample points.

$P\text{(}{S}_1\text{)}=0.1$, and $P\text{(}{S}_2\text{)}=0.2$so,

$P\text{(}{S}_3\text{)}=0.7$

$$\begin{gathered} P\text{(}{S}_1\text{)}+P\text{(}{S}_2\text{)}+P\text{(}{S}_3\text{) = 1} \\ 0.1+0.2+P\left(S_3\right)=1 \\ P\left(S_3\right)=1-0.3 \\ P\left(S_3\right)=0.7 \end{gathered}$$

Hence, $P\left(S_3\right)=0.7$

Find P(A).

b.

If an event$A=\text{{}{S}_2,S_3\text{}}$ , then,

$\begin{array}{rcl}P\text{(}A\text{)}& =& P\text{(}{S}_2\text{)}+P\text{(}{S}_3\text{)}\\ & =& 0.2+0.3\\ & =& 0.5\\ & & \end{array}$

Therefore,$P\left(A\right)=0.5$