Question

A fair six-sided dice is thrown and the scores are noted.Event X: The total of the two scores is 4.Even Y: The first score is 2 or 5.

a)Construct the table by showing the sample spaces.

b)Find the probability of Xand Y.

c)Are events X and Y independent?

Answer 1

(a) A dice is thrown two times

So, total outcomes = $6^2=36$

Event X: The total of the two scores is 4

So, Sample space for X = { (1,3) , (2,2) , (3,1) }

Event Y: The first score is 2 or 5

So, Sample space for Y = { (2,1) , (2,2) , (2,3) , (2,4) , (2,5) , (2,6)

(5,1) , (5,2) , (5,3) , (5,4) , (5,5) , (5,6) }

Table:

(b) Total outcomes for X = 3

So, $P(X)=\dfrac{3}{36}=\dfrac{1}{12}$

Total outcomes for Y = 12

So, $P(Y)=\dfrac{12}{36}=\dfrac{1}{3}$

(c) For two events two be independent:

$P(X)\cdot P(Y)=P(X\cap Y)$

So, $P(X)\cdot P(Y)= \dfrac{1}{12}\times \dfrac{1}{3}=\dfrac{1}{36}$

and From table : $X\cap Y =1$

i.e. only one outcome is common for event X and event Y i.e. (2,2)

So, $P(X\cap Y)=\dfrac{1}{36}$

Hence, $P(X)\cdot P(Y)=P(X\cap Y)$

So, we can say that Event X and Event Y are Independent