Question

Two fair six-sided dice are rolled once. Let (X, Y) denote the pair of outcomes of the two rolls.

a) Find the probability that the two rolls result in the same outcomes.

b) Find the probability that the face of at least one of the dice is 4.

c) Find the probability that the sum of the dice is greater than 6.

d) Given that X less than or equal to 4 find the probability that Y > X.

Answer 1

total number of outcomes on 2 dice =6*6 =36 (Since 6 outcome on each dice)

a)

number of ways both outcomes are same =6 (these are (1,1),(2,2),(3,3) and so on)

P( two rolls result in the same outcomes ) =6/36 =1/6

b)

P(at least one dice shows a 4) =1-P(none shows 4) =1-(5/6)*(5/6) =1-25/36 =11/36

c)

probability that the sum of the dice is greater than 6 =P(S=7)+P(S=8)+P(S=9)+P(S=10)+P(S=11)+P(S=12)

=(6/36)+(5/36)+(4/36)+(3/36)+(2/36)+(1/36) =21/36 =7/12

d)total number of outcome in which X is less than 4 =3*6 =18

number of outcome in which X is less than 4 and Y>X =(5+4+3=12)

P(Y>X |X<4) =12/18 =2/3