past paper q2

[maths_arghh234]

hey so this is one of the other questions i'm completely stuck on i don't know where to start ....

a fair cubical die is thrown twice. let A denote the event that thescore on the first throw is less than the score on the second throw and let B denote the event that the scores on the two throws differ by 1

a) calculat P(A)
b) Calculate P(B)
C) determine whether or not A and B are independent.

thanks

 

[soroban]

[size=`120]Hello, maths_arghh234![/size]

A fair cubical die is thrown twice.
Let \(\displaystyle A\) denote the event that the score on the first throw is less than the score on the second throw.
Let \(\displaystyle B\) denote the event that the scores on the two throws differ by 1

a) Calculate \(\displaystyle P(A).\)

b) Calculate \(\displaystyle P(B) .\)

C) Determine whether or not \(\displaystyle A\) and \(\displaystyle B\) are independent.

With a pair-of=dice problem, you can list the 36 outcomes . . . or just visualize them.



\(\displaystyle a)\;\;x \,<\,y\)

. . \(\displaystyle \begin{array}{|cccccc|}\hline * & (1,2) & (1,3) & (1,4) & (1,5) & (1,6) \\ * &*& (2,3) & (2,4) & (2,5) & (2,6) \\ *&*&*& (3,4) & (3,5) & (3,6) \\ *&*&*&*& (4,5) & (4,6) \\ *&*&*&*&*& (5,6) \\ *&*&*&*&*&* \\ \hline \end{array} \qquad\text{15 outcomes}\)

. . \(\displaystyle \text{Therefore: }\(x < y) \:=\:\frac{15}{36} \:=\:\frac{5}{12}\)


\(\displaystyle b)\;\;|x-y| \:=\:1\)

. . \(\displaystyle \begin{array}{|cccccc|}\hline *& (1,2) &*&*&*&* \\ (2,1) &*& (2,3) &*&*&* \\ *& (3,2) &*& (3,4) &*&* \\ *&*&(4,3) &*& (4,5) &* \\ *&*&*& (5,4) &*& (5,6) \\ *&*&*&*& (6,5) &* \\ \hline \end{array} \qquad\text{10 outcomes}\)

. . \(\displaystyle \text{Therefore: }\(|x-y| = 1) \:=\:\frac{10}{36} \:=\:\frac{5}{18}\)



\(\displaystyle {c)\;\; P(A \cap B) \:=\:\frac{5}{36}\)
. . \(\displaystyle P(A)\cdot P(B) \:=\:\frac{5}{12}\cdot\frac{5}{18} \:=\:\frac{25}{216}\)

\(\displaystyle \text{Events }A\text{ and }B\text{ are }not\text{ independent.}\)

 

[maths_arghh234]

oh wow !! .. it seems obvious now .. thank you !!