Two dice are rolled. let x = 1 if the dice come up doubles and let x = 0 otherwise. let y = 1 if the sum is 6, and let y = 0 otherwise. let z = 1 if the dice come up both doubles and with a sum of 6 (that is, double 3), and let z = 0 otherwise. let px denote the success probability for x. find px. let py denote the success probability for y. find py. let pz denote the success probability for z. find pz. are x and y independent? does pz = pxpy? does z = xy? explain.

pX = 1/6; pY = 5/36; pZ = 1/36.

X and Y are dependent. .

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].

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.

2^y = ln 5, or   y(ln 2) = ln 5.   solving for y,   y = (ln 5) / (ln 2).   y=5

step-by-step explanation:

its the second choice.

step-by-step explanation:

the elements in the range will be 7 less than the elements in the domain.