When two fair dice are
rolled, the outcomes are

11,12,13,14,.....,32,33,34,....64,65,66.

number of outcomes when one die is thrown = 6

number
of outcomes when two dice are thrown = 6² = 36

So no. of
outcomes = 36

a)

outcomes
of event E(sum < 6) = (11,12,13,14,21,22,23,31,32,41)

outcomes
of event F(sum is even) =
(11,13,15,22,24,26,31,33,35,42,44,46,51,53,55,62,64,66)

n(F) = 18

P(F) =
18/36 = 1/2

EΠF =
(11,13,22,31)

n(EΠF) = 4

P(EΠF) =
4/36 = 1/9

P(E|F)
= P(EΠF) / P(F) = (1/9)/(1/2) = (1×2) / ( 1×9) = 2/9

b)

outcomes
of event E(sum = 10) = (46,55,64)

outcomes
of event F(roll is doubles) =
(11,22,33,44,55,66)

n(F) = 6

P(F) =
6/36 = 1/6

EΠF = (55)

n(EΠF) = 1

P(EΠF) =
1/36

P(E|F)
= P(EΠF) / P(F) = (1/36)/(1/6) =(1×6)/(1×36) = 1/6

c)

outcomes
of event E(sum is even) =
(11,13,15,22,24,26,31,33,35,42,44,46,51,53,55,62,64,66)

outcomes
of event F(sum < 6) = (11,12,13,14,21,22,23,31,32,41)

n(F) = 10

P(F) =
10/36 = 5/18

EΠF =
(11,13,22,31)

n(EΠF) = 4

P(EΠF) =
4/36 = 1/9

P(E|F)
= P(EΠF) / P(F) = (1/9)/(5/18) = (1×18) / ( 5×9)
= 2/5

d)

outcomes
of event E(roll is doubles) =
(11,22,33,44,55,66)

outcomes
of event F(sum = 10) = (46,55,64)

n(F) = 3

P(F) =
3/36 = 1/12

EΠF = (55)

n(EΠF) = 1

P(EΠF) =
1/36

P(E|F)
= P(EΠF) / P(F) = (1/36)/(1/12) =(1×12)/(1×36) = 1/3

e)

outcomes
of event E(sum>7) = (26,35,36,44,45,46,53,54,55,56,62,63,64,65,66)

outcomes
of event F(neither die is a 6) =

(11,12,13,14,15,21,22,23,24,25,31,32,33,34,35,41,42,43,44,45,51,52,53,54,5)

n(F) = 25

P(F) =
25/36

EΠF =
(35,44,45,53,54,55)

n(EΠF) = 6

P(EΠF) =
6/36 = 1/6

P(E|F)
= P(EΠF) / P(F) = (1/6)/(25/36) =(1×36)/(6×25) = 6/25

f)

outcomes
of event E(sum is odd) =
(12,14,16,21,23,25,32,34,36,41,43,45,52,54,56,61,63,65)

outcomes
of event F(atleast one die is a 6) = (16,26,36,46,56,61,62,63,64,65,66,)

n(F) = 11

P(F) =
11/36

EΠF =
(16,36,56,61,63,65)

n(EΠF) = 6

P(EΠF) =
6/36 = 1/6

P(E|F)
= P(EΠF) / P(F) = (1/6)/(11/36) =(1×36)/(6×11) = 6/11
g)

outcomes
of event E(neither die is a 6) =

(11,12,13,14,15,21,22,23,24,25,31,32,33,34,35,41,42,43,44,45,51,52,53,54,55)

outcomes
of event F(sum>7) = (26,35,36,44,45,46,53,54,55,56,62,63,64,65,66)

n(F) = 15

P(F) = 15/36 = 5/12

EΠF = (35,44,45,53,54,55)

n(EΠF) = 6

P(EΠF) = 6/36 = 1/6

P(E|F) = P(EΠF) / P(F) = (1/6)/(5/12) =(1×12)/(6×5) = 2/5
h)

outcomes
of event E(atleast one die is a 6) = (16,26,36,46,56,61,62,63,64,65,66,)

outcomes
of event F(sum is odd) =
(12,14,16,21,23,25,32,34,36,41,43,45,52,54,56,61,63,65)

n(F) = 18

P(F) = 18/36 = 1/2

EΠF = (16,36,56,61,63,65)

n(EΠF) = 6

P(EΠF) = 6/36 = 1/6

P(E|F) = P(EΠF) / P(F) = (1/6)/(1/2) =(1×2)/(6×11) = 1/3