Question

Exercise 7-12: Roll a pair of fair 3-sided dice. Let F denote the number of dots on the first die and let T denote the total number of dots. Determine the Cov(F,T).

Answer 1

Given that,

We Roll a pair of fair 3-sided dice. Let F denote the number of dots on the first die and let T denote the total number of dots.

For this experiment sample space is given by,

Sample space= { (1,1), (2,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) }

Now a 3-sided dice has only 3 dots. i.e. F takes the value 1, 2, 3 And T takes the value 2, 3, 4, 5, 6

[ Note: Probability of a certain event E is given by, P(E) = (Number of favorable outcomes for this event/Total number of outcomes) ]

Now joint probability distribution of F and T is given by,

Column wise F\ Row wise T	1	2	3	Total
2	1/9	0	0	1/9
3	1/9	1/9	0	2/9
4	1/9	1/9	1/9	3/9
5	0	1/9	1/9	2/9
6	0	0	1/9	1/9
Total	3/9	3/9	3/9	1

[ Note:- P(T=1, F=2) = P(1st die shown 1 dots and total number of dots are 2) = 1/9

Since, total number of outcomes are = 9, and favorable outcomes for this event is = 1

favorable outcomes for this event is (1,1). In this outcomes 1st die shown one dots and total number of dots are 2.

P(T=1, F=3) = P(1st die shown 1 dots and total number of dots are 3) = 1/9

Since, total number of outcomes are = 9, and favorable outcomes for this event is = 1

favorable outcomes for this event is (1,2). In this outcomes 1st die shown one dots and total number of dots are 3.

P(T=1, F=5) = P(1st die shown 1 dots and total number of dots are 5) = 0

Since, there is no favorable outcomes for this event. Because if 1st die shown 1 dots then highest number of dots we get 3.

P(T=2, F=5) = P(1st die shown 2 dots and total number of dots are 5) = 1/9

Since, total number of outcomes are = 9, and favorable outcomes for this event is = 1

favorable outcomes for this event is (2,3). In this outcomes 1st die shown one dots and total number of dots are 5.

Similarly we get all the other probabilities ]

Now we want to determine the value of Cov(F, T).

Formula:-

  

[ Round to four decimal places ]

Answer:- Cov(F,T) = 8.6667  [ Round to four decimal places ]