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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鈥檛 choose the stated option

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

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

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

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