Appeals of federal civil trials. The Journal of the American Law and Economics Association (Vol. 3, 2001) publishedthe results of a study of appeals of federal civil trials. Thefollowing table, extracted from the article, gives a breakdownof 2,143 civil cases that were appealed by either theplaintiff or the defendant. The outcome of the appeal, aswell as the type of trial (judge or jury), was determined foreach civil case. Suppose one of the 2,143 cases is selected

at random and both the outcome of the appeal and type of trial are observed.

	Jury	Judge	Totals
Plaintiff trial win-reserved	194	71	265
Plaintiff trial win-affirmed/dismissed	429	240	669
Defendant trial win-reserved	111	68	179
Defendant trial win- affirmed/dismissed	731	678	1030
Total	1465	678	2143

a. Find P (A), where A = {jury trial}.

b. Find P (B), where B = {plaintiff trial win is reversed}.

c. Are A and B mutually exclusive events?

d. Find$\mathbf{P}\mathbf{(}{\mathbf{A}}^C\mathbf{)}$

e. Find$\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{)}$

f. Find$\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{)}$

Step by step solution

01

Important formula        

The formula for probability is

$\begin{array}{l}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{P}\mathbf{=}\frac{\mathrm{Favourable}\mathrm{out}\mathrm{comes}}{\mathrm{Total}\mathrm{out}\mathrm{comes}}\\ \mathbf{P}\mathbf{(}{\mathbf{A}}^C\mathbf{)}\mathbf{=}\mathbf{1}-\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\end{array}$

02

(a) Find P(A)

The total of jury trials is 1465.

So, the value of P(A) is

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\frac{1465}{2143}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{684}\end{array}$

So, the value of P(A) is 0.684

03

(b) Find the value of P(B)

The total of plaintiff wins is reserved is 265. So

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{=}\frac{265}{2143}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{124}\end{array}$

Hence, the value of P(B) is 0.124.

04

(c) Are A and B mutually exclusive events

No, A and B are not mutually exclusive events because$\mathbf{P}\mathbf{(}\mathbf{A}\cap \mathbf{B}\mathbf{)}$ is not equal to zero.

05

(d) Find P(AC)

$\begin{array}{l}\mathbf{P}\mathbf{(}{\mathbf{A}}^C\mathbf{)}\mathbf{=}\mathbf{1}-\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}-\mathbf{0}\mathbf{.}\mathbf{684}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{316}\end{array}$

Accordingly, the value of $\mathbf{P}\mathbf{(}{\mathbf{A}}^C\mathbf{)}$is 0.316

06

(e) Find P(A∪B)

$\begin{array}{l}\mathbf{P}\mathbf{(}\mathbf{A}\cup \mathbf{B}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{)}\mathbf{+}\mathbf{P}\mathbf{(}\mathbf{B}\mathbf{)}-\mathbf{P}\mathbf{(}\mathbf{A}\cap \mathbf{B}\mathbf{)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{684}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{124}-\mathbf{0}\mathbf{.}\mathbf{090}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{717}\end{array}$

Therefore, the value of $\mathbf{P}\mathbf{(}\mathbf{A}\cup \mathbf{B}\mathbf{)}$is 0.717.

07

(f) Determine P(A∩B)

$\begin{array}{c}\mathbf{P}\mathbf{(}\mathbf{A}\cap \mathbf{B}\mathbf{)}\mathbf{=}\frac{194}{2143}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{090}\end{array}$

Henceforth, the value of $\mathbf{P}\mathbf{(}\mathbf{A}\cap \mathbf{B}\mathbf{)}$is 0.090.

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