A Payoff Table, the Prior Probabilities for Three States of Nature

question 41

Essay

A payoff table, the prior probabilities for three states of nature and the likelihood probabilities are shown below.
Payoff Table: $\begin{array}{ll}&\text { Alternative }\\\text { State of Nature }&\quad a_1\quad\quad a_2\quad\quad a_3\\\begin{array}{ll}s_1\\s_2\\s_3\end{array}&\begin{array}{|lll|}\hline80\quad&120\quad&9\quad \\60&130&170\\200&140&100\\\hline\end{array}\end{array}$
Prior Probabilities:
P( $s _ { 1 }$ ) = 0.4, P( $S _ { 2 }$ ) = 0.5, P( $S _ { 3 }$ ) = 0.1.
Likelihood Probabilities: $\begin{array}{ll}&\quad I_1\quad\quad I_2\quad\quad I_3\\\begin{array}{ll}s_1\\s_2\\s_3\end{array}&\begin{array}{|lll|}\hline0.5\quad&0.3\quad&0.2\quad \\0.2&0.6&0.2\\0.1&0.2&0.7\\\hline\end{array}\end{array}$
A) Use the prior and likelihood probabilities to calculate the posterior probabilities for the experimental outcome $I _ { 1 }$ .
B) Use the posterior probabilities from a. to recalculate the expected monetary value of each act, then determine the optimal act and the EMV*.
C) Use the prior and likelihood probabilities to calculate the posterior probabilities for the experimental outcome $I _ { 2 }$ .
D) Use the posterior probabilities from c. to recalculate the expected monetary value of each act, then determine the optimal act and the EMV*.
E) Use the prior and likelihood probabilities to calculate the posterior probabilities for the experimental outcome $I _ { 3 }$ .
F) Use the posterior probabilities from e. to recalculate the expected monetary value of each act, then determine the optimal act and the EMV*.
G) Use your answers to parts a. to f. to calculate the expected monetary value with additional information.
H) Calculate the expected value of sample information.

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Definitions:

False Negative

An error in test results in which a positive condition is incorrectly reported as negative.

Discrete Probability Distribution

A statistical distribution that shows the probabilities of outcomes with distinct values.

Standard Deviation

A metric that quantifies the spread or variation of a data collection in comparison to its average value.

Probability Distribution

Describes how probabilities are distributed over the values of a random variable; it's a mathematical function showing the likelihood of different outcomes.