Guilt in decision making.Refer to the Journal of Behavioral Decision Making(January 2007) study of theeffect of guilt emotion on how a decision maker focuseson a problem, Exercise 3.48 (p. 183). The results (numberresponding in each category) for the 171 study participantsare reproduced in the table below. Suppose one of the 171participants is selected at random.

Emotional State	Choose Stated Option	Do Not Choose Stated Option	Totals
Guilt Anger Neutral	45 8 7	12 50 49	57 58 56
Totals	60	111	171

a.Given that the respondent is assigned to the guilty state, what is the probability that the respondent chooses the stated option?

b.If the respondent does not choose to repair the car, what is the probability that the respondent is in the anger state?

c.Are the events {repair the car} and {guilty state }
independent?

P(A): Guilty respondents who choose the stated option

P(B): Total respondents in a guilty state

To find the probability (P) that the chosen participant chooses the stated option,

$\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right|}\mathit{B}\mathbf{)}\mathbf{=}\frac{\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}}{\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}}\mathbf{=}\frac{\mathbf{45}}{\mathbf{57}}$

Therefore, the probability of picking a respondent who chooses the stated option given that he is in a guilty state is 45/57.

P(C): Respondents who do not choose the stated option

P(D): Respondents don’t choose the stated option

$\mathit{P}\mathbf{\left(}\mathit{C}\mathbf{\right|}\mathit{D}\mathbf{)}\mathbf{=}\frac{\mathbf{P}\mathbf{(}\mathbf{C}\mathbf{\cap }\mathbf{D}\mathbf{)}}{\mathbf{P}\mathbf{\left(}\mathbf{D}\mathbf{\right)}}\mathbf{=}\frac{\frac{\mathbf{50}}{\mathbf{171}}}{\frac{\mathbf{111}}{\mathbf{171}}}\mathbf{=}\frac{\mathbf{50}}{\mathbf{111}}$

The respondent's probability of not choosing to repair the car and being angry is 50/111.

Both events, repairing the car and feeling guilty, will be independent if the occurrence of feeling guilty does not affect the occurrence of repairing the car.

P(A) = Respondents choosing to repair the car

P(B) = Respondents feeling guilty

$\begin{array}{rcl}\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right|}\mathit{B}\mathbf{)}& \mathbf{=}& \mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\\ \frac{\mathbf{60}}{\mathbf{57}}& \mathbf{=}& \frac{\mathbf{57}}{\mathbf{171}}\\ \mathit{B}\mathit{u}\mathit{t}\mathbf{,}\frac{\mathbf{60}}{\mathbf{57}}& \mathbf{\ne }& \frac{\mathbf{57}}{\mathbf{171}}\\ & & \end{array}$

Therefore, repairing the car and feeling guilty are not independent events.