Answer: pX = 1/6; pY = 5/36; pZ = 1/36.
X and Y are dependent. [tex]pZ\neq pXpY[/tex] .
Z = XY
The two events X and Y are dependent hence for Z to occur both Y and X must occur.
Step-by-step explanation:
First thing to know is, Probability of an event = the total possible occurrences of an event/the total sample size
The total sample size of two dice is 36.
If X =1 when the dice come up as doubles, then the total possible occurrences of doubles (11,22,33,44,55,66) are 6. So, the probability of X occurring is
pX = 6/36 = 1/6
If Y =1 when the dice sum is 6. then the total possible occurrences of a sum of six (15,51,33,24,42) are 5. So, the probability of Y occurring is
pY = 5/36
If Z =1 when the dice are both doubles and a sum of 6, then the total possible occurrences of both doubles and a sum of 6 (33) is 1. So, the probability of Z occurring is
pZ = 1/36
X and Y are dependent as can be proven by the next step,
The occurrence of event Z happens when event X and event Y occur so a multiplication of pX and pY would equate to pZ if the two events are mutually exclusive (independent),
pXpY = 1/6*5/36= 5/216 which is not equal to pZ (1/36) [THIS WAS ALREADY ESTABLISHED].
[tex]pZ\neq pXpY[/tex]
However, the success of X and the success of Y is multiplied,the success of Z is established. However, when either X or Y does not occur, Z is not possible.
Z=1 ; (when X=1 and Y=1)
Z=0; (when either X=0 or Y=0 or when both =0)
So Z = XY
The explanation for this is that Z is dependent on two mutually dependent occurrences to occur and that X and Y are mutually dependent.
