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In a simultaneous throw of a pair of dice, find the probability of getting:

(i) 8 as the sum

(ii) a doublet

(iii) a doublet of prime numbers

(iv) a doublet of odd numbers

(v) a sum greater than9

(vi) An even number on first

(vii) an even number on one and a multiple of 3 on the other

(viii) neither 9 nor 11 as the sum of the numbers on the faces

(ix) a sum less than 6

(x) a sum less than 7

(xi) a sum more than 7

(xii) at least once a number other than 5 on any dice.

Answer:

Let us construct a table.

Here the first number denotes the outcome of first die and second number denotes the outcome of second die.

(i) 8 as the sum

Total number of outcomes in the above table are 36

Number of outcomes having 8 as sum are: (6, 2), (5, 3), (4, 4), (3, 5) and (2, 6) Therefore numbers of outcomes having 8 as sum are 5

Probability of getting numbers of outcomes having 8 as sum is = Total numbers/Total number of outcomes

= 5/36

∴ Probability of getting numbers of outcomes having 8 as sum is 5/36

(ii) a doublet

Total number of outcomes in the above table are 36

Number of outcomes as doublet are: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6) Number of outcomes as doublet are 6

Probability of getting numbers of outcomes as doublet is = Total numbers/Total number of outcomes

= 6/36

= 1/6

∴ Probability of getting numbers of outcomes as doublet is 1/6

(iii) a doublet of prime numbers

Total number of outcomes in the above table are 36

Number of outcomes as doublet of prime numbers are: (1, 1), (3, 3), (5, 5) Number of outcomes as doublet of prime numbers are 3

Probability of getting numbers of outcomes as doublet of prime numbers is = Total numbers/Total number of outcomes

= 3/36

= 1/12

∴ Probability of getting numbers of outcomes as doublet of prime numbers is 1/12

(iv) a doublet of odd numbers

Total number of outcomes in the above table are 36

Number of outcomes as doublet of odd numbers are: (1, 1), (3, 3), (5, 5) Number of outcomes as doublet of odd numbers are 3

Probability of getting numbers of outcomes as doublet of odd numbers is = Total numbers/Total number of outcomes

= 3/36

= 1/12

∴ Probability of getting numbers of outcomes as doublet of odd numbers is 1/12

(v) a sum greater than 9

Total number of outcomes in the above table are 36

Number of outcomes having sum greater than 9 are: (4, 6), (5, 5), (5, 6), (6, 6), (6, 4), (6, 5)

Number of outcomes having sum greater than 9 are 6

Probability of getting numbers of outcomes having sum greater than 9 is = Total numbers/Total number of outcomes

= 6/36 = 1/6

∴ Probability of getting numbers of outcomes having sum greater than 9 is 1/6

(vi) An even number on first

Total number of outcomes in the above table are 36

Number of outcomes having an even number on first are: (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5) and (6, 6)

Number of outcomes having an even number on first are 18

Probability of getting numbers of outcomes having an even number on first is = Total numbers/Total number of outcomes

= 18/36

= 1/2

∴ Probability of getting numbers of outcomes having an even number on first is 1/2

(vii) An even number on one and a multiple of 3 on the other Total number of outcomes in the above table are 36

Number of outcomes having an even number on one and a multiple of 3 on the other are: (2, 3), (2, 6), (4, 3), (4, 6), (6, 3) and (6, 6)

Number of outcomes having an even number on one and a multiple of 3 on the other are 6

Probability of getting an even number on one and a multiple of 3 on the other is = Total numbers/Total number of outcomes

= 6/36

= 1/6

∴ Probability of getting an even number on one and a multiple of 3 on the other is 1/6

(viii) Neither 9 nor 11 as the sum of the numbers on the faces Total number of outcomes in the above table are 36

Number of outcomes having 9 nor 11 as the sum of the numbers on the faces are: (3, 6), (4, 5), (5, 4), (5, 6), (6, 3) and (6, 5)

Number of outcomes having neither 9 nor 11 as the sum of the numbers on the faces are 6 Probability of getting 9 nor 11 as the sum of the numbers on the faces is = Total numbers/Total number of outcomes

= 6/36

= 1/6

Probability of outcomes having 9 nor 11 as the sum of the numbers on the faces P (E) = 1/6

∴Probability
of getting neither 9 nor 11 as the sum of the numbers on the faces is 1/6
Probability of outcomes not having 9
nor 11 as the sum of the numbers on the faces is

given by P (E) =
1 – 1/6 = (6-1)/5 = 5/6

∴ Probability of outcomes not having 9 nor 11 as the sum of the numbers on the faces is 5/6

(ix) A sum less than 6

Total number of outcomes in the above table are 36

Number of outcomes having a sum less than 6 are: (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)

Number of outcomes having a sum less than 6 are 10

Probability of getting a sum less than 6 is = Total numbers/Total number of outcomes

= 10/36

= 5/18

∴ Probability of getting sum less than 6 is 5/18

(x) A sum less than 7

Total number of outcomes in the above table are 36

Number of outcomes having a sum less than 7 are: (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (5, 1)

Number of outcomes having a sum less than 7 are 15

Probability of getting a sum less than 7 is = Total numbers/Total number of outcomes

= 15/36

= 5/12

∴ Probability of getting sum less than 7 is 5/12

(xi) A sum more than 7

Total number of outcomes in the above table are 36

Number of outcomes having a sum more than 7 are: (2, 6), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6), (5, 3), (5, 4), (5, 5), (5, 6), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

Number of outcomes having a sum more than 7 are 15

Probability of getting a sum more than 7 is = Total numbers/Total number of outcomes

= 15/36

= 5/12

∴ Probability of getting sum more than 7 is 5/12

(xii) At least once

Total number of outcomes in the above table 1 are 36 Number of outcomes for at least once are 11

Probability of getting outcomes for at least once is = Total numbers/Total number of outcomes

= 11/36

∴ Probability of getting outcomes for at least once is 11/36

(xiii) A number other than 5 on any dice.

Total number of outcomes in the above table 1 are 36

Number of outcomes having 5 on any die are: (1, 5), (2, 5), (3, 5), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 5)

Number of outcomes having outcomes having 5 on any die are 15

Probability of getting 5 on any die is = Total numbers/Total number of outcomes

= 11/36

∴ Probability of getting 5 on any die is 11/36 Probability of not getting 5 on any die P (E) = 1 – P (E)

= 1 – 11/36

= (36-11)/36

= 25/36

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