Chapter 4.6, Problem 4.65E

To determine

(a)

The value of $P\left(A\right)$.

Answer to Problem 4.65E

  $P\left(A\right)=\frac{154}{256}$

Explanation of Solution

Given:

The classification of 256 soccer players is shown by the following table.

Severity	Practice (P)	Game (G)	Total
Minor (A)	66	88	154
Moderate (B)	23	44	67
Major (C)	12	23	35
Total	101	155	256

Formula Used:

Probability of an event is given by

  $\frac{\text{Number}\text{of}\text{Favorable}\text{Outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Calculation:

Let T be the event of total players.

  $n\left(T\right)=256$

  $A$ be the event of choosing a minor.

  $n\left(A\right)=154$

The value of $P\left(A\right)$ can be calculated as

  $\begin{array}{c}P\left(A\right)=\frac{n\left(A\right)}{n\left(T\right)}\\ =\frac{154}{256}\end{array}$

To determine

(b)

The value of $P\left(G\right)$.

Answer to Problem 4.65E

  $P\left(G\right)=\frac{155}{256}$

Explanation of Solution

Given:

The classification of 256 soccer players is shown by the following table.

Severity	Practice (P)	Game (G)	Total
Minor (A)	66	88	154
Moderate (B)	23	44	67
Major (C)	12	23	35
Total	101	155	256

Formula Used:

Probability of an event is given by

  $\frac{\text{Number}\text{of}\text{Favorable}\text{Outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Calculation:

Let T be the event of total players.

  $n\left(T\right)=256$

  $G$ be the event of choosing a person playing the Game.

  $n\left(G\right)=155$

The value of $P\left(G\right)$ can be calculated as

  $\begin{array}{c}P\left(G\right)=\frac{n\left(G\right)}{n\left(T\right)}\\ =\frac{155}{256}\end{array}$

To determine

(c)

The value of $P\left(A\cap G\right)$.

Answer to Problem 4.65E

  $P\left(A\cap G\right)=\frac{88}{256}$

Explanation of Solution

Given:

The classification of 256 soccer players is shown by the following table.

Severity	Practice (P)	Game (G)	Total
Minor (A)	66	88	154
Moderate (B)	23	44	67
Major (C)	12	23	35
Total	101	155	256

Formula Used:

Probability of an event is given by

  $\frac{\text{Number}\text{of}\text{Favorable}\text{Outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Calculation:

Let T be the event of total players.

  $n\left(T\right)=256$

  $A\cap G$ be the event of choosing a Minor who playing the game.

  $n\left(A\cap G\right)=88$

The value of $P\left(A\cap G\right)$ can be calculated as

  $\begin{array}{c}P\left(A\cap G\right)=\frac{n\left(A\cap G\right)}{n\left(T\right)}\\ =\frac{88}{256}\end{array}$

To determine

(d)

The value of $P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)$.

Answer to Problem 4.65E

  $P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)=\frac{88}{154}$

Explanation of Solution

Given:

The classification of 256 soccer players is shown by the following table.

Severity	Practice (P)	Game (G)	Total
Minor (A)	66	88	154
Moderate (B)	23	44	67
Major (C)	12	23	35
Total	101	155	256

Formula Used:

For any two events $A$ and B,

  $P\left(B/A\right)=\frac{P\left(A\text{and}\text{}B\right)}{P\left(A\right)}$

Where

  $P\left(B/A\right)=$ The probability of occurring $B$ under the assumption that $A$ has been occurred.

Calculation:

  $A$ be the event of choosing a minor.

  $P\left(A\right)=\frac{154}{256}$

  $A\cap G$ be the event of choosing a Minor who are playing the game.

  $P\left(A\cap G\right)=\frac{88}{256}$

The value of $P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)$ can be calculated as

  $\begin{array}{c}P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)=\frac{P\left(A\cap G\right)}{P\left(A\right)}\\ =\frac{\frac{88}{256}}{\frac{154}{256}}\\ =\frac{88}{154}\end{array}$

To determine

(e)

The value of $P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$B$}\right.\right)$.

Answer to Problem 4.65E

  $P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$B$}\right.\right)=\frac{44}{67}$

Explanation of Solution

Given:

The classification of 256 soccer players is shown by the following table.

Severity	Practice (P)	Game (G)	Total
Minor (A)	66	88	154
Moderate (B)	23	44	67
Major (C)	12	23	35
Total	101	155	256

Formula Used:

For any two events $A$ and B,

  $P\left(B/A\right)=\frac{P\left(A\text{and}\text{}B\right)}{P\left(A\right)}$

Where

  $P\left(B/A\right)=$ The probability of occurring $B$ under the assumption that $A$ has been occurred.

Calculation:

Let T be the event of total players.

  $n\left(T\right)=256$

  $B$ be the event of choosing a Moderate.

  $n\left(B\right)=67$

The value of $P\left(B\right)$ can be calculated as

  $\begin{array}{c}P\left(B\right)=\frac{n\left(B\right)}{n\left(T\right)}\\ =\frac{67}{256}\end{array}$

  $B\cap G$ be the event of choosing a Modarate who are playing the game.

  $P\left(B\cap G\right)=\frac{44}{256}$

The value of $P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$B$}\right.\right)$ can be calculated as

  $\begin{array}{c}P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$B$}\right.\right)=\frac{P\left(B\cap G\right)}{P\left(B\right)}\\ =\frac{\frac{44}{256}}{\frac{67}{256}}\\ =\frac{44}{67}\end{array}$

To determine

(f)

The value of $P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$C$}\right.\right)$.

Answer to Problem 4.65E

  $P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$C$}\right.\right)=\frac{23}{35}$

Explanation of Solution

Given:

The classification of 256 soccer players is shown by the following table.

Severity	Practice (P)	Game (G)	Total
Minor (A)	66	88	154
Moderate (B)	23	44	67
Major (C)	12	23	35
Total	101	155	256

Formula Used:

For any two events $A$ and B,

  $P\left(B/A\right)=\frac{P\left(A\text{and}\text{}B\right)}{P\left(A\right)}$

Where

  $P\left(B/A\right)=$ The probability of occurring $B$ under the assumption that $A$ has been occurred.

Calculation:

Let T be the event of total players.

  $n\left(T\right)=256$

  $C$ be the event of choosing a Major.

  $n\left(C\right)=35$

The value of $P\left(C\right)$ can be calculated as

  $\begin{array}{c}P\left(C\right)=\frac{n\left(C\right)}{n\left(T\right)}\\ =\frac{35}{256}\end{array}$

  $C\cap G$ be the event of choosing a Major who are playing the game.

  $P\left(C\cap G\right)=\frac{23}{256}$

The value of $P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$C$}\right.\right)$ can be calculated as

  $\begin{array}{c}P\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$C$}\right.\right)=\frac{P\left(C\cap G\right)}{P\left(C\right)}\\ =\frac{\frac{23}{256}}{\frac{35}{256}}\\ =\frac{23}{35}\end{array}$

To determine

(g)

The value of $P\left(\raisebox{1ex}{$C$}\!\left/ \!\raisebox{-1ex}{$P$}\right.\right)$.

Answer to Problem 4.65E

  $P\left(\raisebox{1ex}{$C$}\!\left/ \!\raisebox{-1ex}{$P$}\right.\right)=\frac{12}{101}$

Explanation of Solution

Given:

The classification of 256 soccer players is shown by the following table.

Severity	Practice (P)	Game (G)	Total
Minor (A)	66	88	154
Moderate (B)	23	44	67
Major (C)	12	23	35
Total	101	155	256

Formula Used:

For any two events $A$ and B,

  $P\left(B/A\right)=\frac{P\left(A\text{and}\text{}B\right)}{P\left(A\right)}$

Where

  $P\left(B/A\right)=$ The probability of occurring $B$ under the assumption that $A$ has been occurred.

Calculation:

Let T be the event of total players.

  $n\left(T\right)=256$

  $P$ be the event of choosing an individual doing practice.

  $n\left(P\right)=101$

The value of $P\left(P\right)$ can be calculated as

  $\begin{array}{c}P\left(P\right)=\frac{n\left(P\right)}{n\left(T\right)}\\ =\frac{101}{256}\end{array}$

  $C$ be the event of choosing a Major.

  $C\cap P$ be the event of choosing a Major who are doing practice.

  $P\left(C\cap P\right)=\frac{12}{256}$

The value of $P\left(\raisebox{1ex}{$C$}\!\left/ \!\raisebox{-1ex}{$P$}\right.\right)$ can be calculated as

  $\begin{array}{c}P\left(\raisebox{1ex}{$C$}\!\left/ \!\raisebox{-1ex}{$P$}\right.\right)=\frac{P\left(C\cap P\right)}{P\left(P\right)}\\ =\frac{\frac{12}{256}}{\frac{101}{256}}\\ =\frac{12}{101}\end{array}$

To determine

(h)

The value of $P\left(B^c\right)$.

Answer to Problem 4.65E

  $P\left(B^c\right)=\frac{189}{256}$

Explanation of Solution

Given:

The classification of 256 soccer players is shown by the following table.

Severity	Practice (P)	Game (G)	Total
Minor (A)	66	88	154
Moderate (B)	23	44	67
Major (C)	12	23	35
Total	101	155	256

Formula Used:

Probability of an event is given by

  $\frac{\text{Number}\text{of}\text{Favorable}\text{Outcome}}{\text{Number}\text{of}\text{total}\text{outcome}}$

Calculation:

Let T be the event of total players.

  $n\left(T\right)=256$

  $B$ be the event of choosing a Moderate.

  $n\left(B\right)=67$

The value of $P\left(B\right)$ can be calculated as

  $\begin{array}{c}P\left(B\right)=\frac{n\left(B\right)}{n\left(T\right)}\\ =\frac{67}{256}\end{array}$

Therefore, $B^c$ be the event of not o curing B.

  $\begin{array}{c}P\left(B^c\right)=1-P\left(B\right)\\ =1-\frac{67}{256}\\ =\frac{256-67}{256}\\ =\frac{189}{256}\end{array}$