Two indistinguishable dice are rolled; both numbers are prime. (A positive integ

Question

Two indistinguishable dice are rolled; both numbers are prime. (A positive integer is prime if it is neither 1 nor a product of smaller integers.)

Which of the following sets of elements are included in the sample space? (Select all that apply.)

(4,4) (4,5) (4,6)

(5,5) (5,6)

(3,3) (3,4) (3,5) (3,6)

(2,2) (2,3) (2,4) (2,5) (2,6) (2,7)

(5,5) (5,6) (5,7)

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7)

(3,3) (3,4) (3,5) (3,6) (3,7)

(4,4) (4,5) (4,6) (4,7)

(6,6)

(7,7)

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

(6,6) (6,7)

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

Explanation / Answer

now suppose that the dice are indistinguishable.then this mean that we will treat a roll thatresults in(i,j) to be the same as a roll that results in(j,i).so the possible outcome are now:

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

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

(3,3),(3,4),(3,5),(3,6),

(4,4),(4,5),(4,6),

(5,5),(5,6),

(6,6)

however, in this case ,not all outcomes have the same probabiity.the outcomes that are of the form (i,j) will have the probability 1/36;all other outcomes have probability 2/36=1/18.

for example:lets take outcome (3,5).even a roll (3,5)and a roll (5,3) are indistinguisable and so lead the outcome are same.therefore,the probabiity of outcome (3,5) is 2/36=1/18.

now,let compute the probability that the sumof two numbers is equal to 7 when we roll two dice:

in that case that dice are distinguishable,this probability is equal to the probability of getting (1,6)+probabiity of(6,1)=1/36+1/36=1/18.

in the case that the dice are indistinguishable,this probability is equal to the probability of getting(1,6)=1/18.

which has the same answer.

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