Chapter 7, Problem 7.79RE

(a)

To determine

To construct: a diagram the sample space illustrating the possible outcomes on a single turn.

Explanation of Solution

Given:

Rolling a fair eight sided dice by teacher and the student rolling a fair six-sided dice

Winners would be the person with highest numbers and if tie is there than no winner

Calculation:

Suppose

T = Teacher wins round

S = Student wins round

N = No winner

8-sided dice has possible outcomes = 1, 2, 3, 4, 5, 6, 7, 8

6-sided dice has possible outcomes = 1, 2, 3, 4, 5, 6

Teacher
		1	2	3	4	5	6	7	8
Student	1	N	T	T	T	T	T	T	T
	2	S	N	T	T	T	T	T	T
	3	S	S	N	T	T	T	T	T
	4	S	S	S	N	T	T	T	T
	5	S	S	S	S	N	T	T	T
	6	S	S	S	S	S	N	T	T

(b)

To determine

To Calculate: the $\text{P}\left(\text{A}\right)$ .

Answer to Problem 7.79RE

  $\text{P}\left(\text{A}\right)\text{=0}\text{.5625}$

Explanation of Solution

Formula used:

  $\text{Probability}\text{=}\frac{\text{Favourable Cases}}{\text{Total}\text{Cases}}$

Calculation:

Assuming that A= teacher wins the first turn of the game.

By seeing the diagram, it is observed that there are 48 possible cases, while 27 of the 48 cases result in the teach T wining

  $\begin{array}{c}\text{Probability}\text{=}\frac{\text{Favourable Cases}}{\text{Total}\text{Cases}}\\ \text{P}\left(\text{A}\right)\text{=}\frac{\text{27}}{\text{48}}\\ \text{=0}\text{.5625}\end{array}$

(c)

To determine

To Calculate: the $\text{P}\left(\text{A}\cup \text{B}\right)$

Answer to Problem 7.79RE

  $\text{P}\left(A\cup B\right)=0.625$

Explanation of Solution

Formula used:

  $\text{Probability}\text{=}\frac{\text{Favourable Cases}}{\text{Total}\text{Cases}}$

Calculation:

By seeing the diagram, it is observed that there are 48 possible cases, while 8 of the 48 cases result in the 3 for the student.

  $\begin{array}{c}\text{Probability}\text{=}\frac{\text{Favourable Cases}}{\text{Total}\text{Cases}}\\ \text{P}\left(B\right)\text{=}\frac{8}{\text{48}}\\ \text{=0}\text{.1667}\end{array}$

  $\text{P}\left(A\cap B\right)=\frac{5}{48}$

General addition rule

  $\begin{array}{c}\text{P}\left(A\cup B\right)=\text{P}\left(B\right)+\text{P}\left(B\right)-\text{P}\left(A\cap B\right)\\ =\frac{27}{48}+\frac{8}{48}-\frac{5}{48}\\ =\frac{5}{8}\\ =0.625\end{array}$

(d)

To determine

To Explain: that the events A and B are independent, justify answer.

Answer to Problem 7.79RE

Not Independent

Explanation of Solution

Formula used:

Conditional Probability

  $P\left(A|B\right)=\frac{\text{P}\left(B\cap A\right)}{P\left(B\right)}$

Calculation:

  $\begin{array}{c}P\left(B|A\right)=\frac{\text{P}\left(A\cap B\right)}{P\left(A\right)}\\ =\frac{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$48$}\right.}{\raisebox{1ex}{$27$}\!\left/ \!\raisebox{-1ex}{$48$}\right.}\\ =\frac{5}{27}\\ =0.1852\end{array}$

It is observed that the $P\left(B|A\right)\ne P\left(B\right)$ it means the event A and B are not independent which is looking forth.