Question

instructions. show all your work to get full credit. circle your final answers. the exam is worth 40 points that is 20% of your final grade. 1) (4 points) suppose given the sample space s = {e1,e2,e3}. indicate the cases where we have an acceptable assignment of probabilities to the simple events. explain the reasons if an assignment is unacceptable. (a) p(e1) = 1/4; p(e2) = 1/3; p(e3) = 1/4. (b) p(e1) = 1/3; p(e2) = -1/3; p(e3) = 1. (c) p(e1) = 1/5; p(e2) = 3/5; p(e3) = 1/5. (d) p(e1) = 1/2; p(e2) = 0; p(e3) = 1/2.

Explanation:

Step1: Recall probability axioms

The sum of probabilities of all simple - events in a sample space must be equal to 1, and the probability of any event \(P(E)\) must satisfy \(0\leq P(E)\leq1\).

Step2: Analyze part (a)

For the sample space \(S = \{e_1,e_2,e_3\}\), \(P(e_1)=\frac{1}{4}\), \(P(e_2)=\frac{1}{3}\), \(P(e_3)=\frac{1}{4}\). Calculate the sum \(P(e_1)+P(e_2)+P(e_3)=\frac{1}{4}+\frac{1}{3}+\frac{1}{4}=\frac{3 + 4+3}{12}=\frac{10}{12}=\frac{5}{6}
eq1\). So, this is not a valid assignment of probabilities.

Step3: Analyze part (b)

\(P(e_1)=\frac{1}{3}\), \(P(e_2)=-\frac{1}{3}\), \(P(e_3) = 1\). Since \(P(e_2)=-\frac{1}{3}<0\), this is not a valid assignment of probabilities because probabilities must be non - negative.

Step4: Analyze part (c)

\(P(e_1)=\frac{1}{5}\), \(P(e_2)=\frac{3}{5}\), \(P(e_3)=\frac{1}{5}\). Calculate the sum \(P(e_1)+P(e_2)+P(e_3)=\frac{1 + 3+1}{5}=1\), and \(0\leq P(e_i)\leq1\) for \(i = 1,2,3\). So, this is a valid assignment of probabilities.

Step5: Analyze part (d)

\(P(e_1)=\frac{1}{2}\), \(P(e_2)=0\), \(P(e_3)=\frac{1}{2}\). Calculate the sum \(P(e_1)+P(e_2)+P(e_3)=\frac{1}{2}+0+\frac{1}{2}=1\), and \(0\leq P(e_i)\leq1\) for \(i = 1,2,3\). So, this is a valid assignment of probabilities.

Answer:

(a) Not a valid probability assignment because \(P(e_1)+P(e_2)+P(e_3)
eq1\).
(b) Not a valid probability assignment because \(P(e_2)<0\).
(c) Valid probability assignment as \(P(e_1)+P(e_2)+P(e_3)=1\) and \(0\leq P(e_i)\leq1\) for \(i = 1,2,3\).
(d) Valid probability assignment as \(P(e_1)+P(e_2)+P(e_3)=1\) and \(0\leq P(e_i)\leq1\) for \(i = 1,2,3\).