Probability because that rolling two dice v the six sided dotssuch together 1, 2, 3, 4, 5 and also 6 dots in each die.

You are watching: A pair of dice is rolled. find the probability of rolling When 2 dice are thrown simultaneously, thus variety of event can be 62 = 36 due to the fact that each die has actually 1 to 6 number top top its faces. Then the feasible outcomes are shown in the below table. Note:

(i) The outcomes (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6) are called doublets.

(ii) The pair (1, 2) and also (2, 1) are various outcomes.

Worked-out problems involving probability because that rolling two dice:

1. 2 dice space rolled. Allow A, B, C be the events of gaining a amount of 2, a sum of 3 and a amount of 4 respectively. Then, present that

(i) A is a simple event

(ii) B and also C space compound events

(iii) A and B space mutually exclusive

Solution:

Clearly, us haveA = (1, 1), B = (1, 2), (2, 1) and also C = (1, 3), (3, 1), (2, 2).

(i) due to the fact that A consists of a solitary sample point, that is a simple event.

(ii) because both B and C contain more than one sample point, each among them is a compound event.

(iii) since A ∩ B = ∅, A and B space mutually exclusive.

2. two dice space rolled. A is the event that the sum of the numbers shown on the 2 dice is 5, and B is the event that at least one the the dice shows up a 3. Are the two occasions (i) mutually exclusive, (ii) exhaustive? Give arguments in assistance of your answer.

Solution:

When two dice room rolled, we have actually n(S) = (6 × 6) = 36.

Now, A = (1, 4), (2, 3), (4, 1), (3, 2), and also

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

(i) A ∩ B = (2, 3), (3, 2) ≠ ∅.

Hence, A and also B space not mutually exclusive.

(ii) Also, A ∪ B ≠ S.

Therefore, A and also B room not exhaustive events.

More examples related come the questions on the probabilities because that throwing 2 dice.

3. 2 dice room thrown simultaneously. Uncover the probability of:

(i) getting six together a product

(ii) obtaining sum ≤ 3

(iii) obtaining sum ≤ 10

(iv) getting a doublet

(v) obtaining a sum of 8

(vi) getting sum divisible by 5

(vii) gaining sum the atleast 11

(viii) getting a many of 3 as the sum

(ix) acquiring a total of atleast 10

(x) gaining an even number together the sum

(xi) gaining a element number together the sum

(xii) gaining a double of even numbers

(xiii) obtaining a multiple of 2 top top one die and a lot of of 3 top top the other die

Solution:

Two various dice room thrown simultaneously being number 1, 2, 3, 4, 5 and 6 on your faces. We understand that in a single thrown the two various dice, the total number of possible outcomes is (6 × 6) = 36.

(i) acquiring six together a product:

Let E1 = event of gaining six as a product. The number whose product is 6 will be E1 = <(1, 6), (2, 3), (3, 2), (6, 1)> = 4

Therefore, probability ofgetting ‘six as a product’

number of favorable outcomesP(E1) = Total variety of possible result = 4/36 = 1/9

(ii) obtaining sum ≤ 3:

Let E2 = event of getting sum ≤ 3. The number whose sum ≤ 3 will certainly be E2 = <(1, 1), (1, 2), (2, 1)> = 3

Therefore, probability ofgetting ‘sum ≤ 3’

number of favorable outcomesP(E2) = Total number of possible outcome = 3/36 = 1/12

(iii) obtaining sum ≤ 10:

Let E3 = occasion of gaining sum ≤ 10. The number whose sum ≤ 10 will certainly be E3 =

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

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

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

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

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

(6, 1), (6, 2), (6, 3), (6, 4)> = 33

Therefore, probability ofgetting ‘sum ≤ 10’

variety of favorable outcomesP(E3) = Total variety of possible outcome = 33/36 = 11/12(iv)getting a doublet:Let E4 = event of obtaining a doublet. The number which doublet will certainly be E4 = <(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)> = 6

Therefore, probability ofgetting ‘a doublet’

number of favorable outcomesP(E4) = Total variety of possible outcome = 6/36 = 1/6

(v)getting a sum of 8:

Let E5 = event of getting a sum of 8. The number i m sorry is a amount of 8 will be E5 = <(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)> = 5

Therefore, probability ofgetting ‘a amount of 8’

variety of favorable outcomesP(E5) = Total number of possible result = 5/36

(vi)getting amount divisible by 5:

Let E6 = event of gaining sum divisible by 5. The number whose sum divisible by 5 will certainly be E6 = <(1, 4), (2, 3), (3, 2), (4, 1), (4, 6), (5, 5), (6, 4)> = 7

Therefore, probability ofgetting ‘sum divisible by 5’

variety of favorable outcomesP(E6) = Total variety of possible result = 7/36

(vii)getting amount of atleast 11:

Let E7 = occasion of obtaining sum that atleast 11. The occasions of the sum of atleast 11 will be E7 = <(5, 6), (6, 5), (6, 6)> = 3

Therefore, probability ofgetting ‘sum of atleast 11’

variety of favorable outcomesP(E7) = Total number of possible result = 3/36 = 1/12

(viii) gaining amultiple the 3 together the sum:

Let E8 = event of getting a many of 3 as the sum. The occasions of a lot of of 3 as the sum will be E8 = <(1, 2), (1, 5), (2, 1), (2, 4), (3, 3), (3, 6), (4, 2), (4, 5), (5, 1), (5, 4), (6, 3) (6, 6)> = 12

Therefore, probability ofgetting ‘a lot of of 3 together the sum’

variety of favorable outcomesP(E8) = Total number of possible result = 12/36 = 1/3

(ix) acquiring a totalof atleast 10:

Let E9 = occasion of obtaining a complete of atleast 10. The events of a full of atleast 10 will be E9 = <(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)> = 6

Therefore, probability ofgetting ‘a total of atleast 10’

number of favorable outcomesP(E9) = Total number of possible outcome = 6/36 = 1/6

(x) acquiring an evennumber as the sum:

Let E10 = event of getting an even number together the sum. The events of an also number as the sum will it is in E10 = <(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 3), (3, 1), (3, 5), (4, 4), (4, 2), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)> = 18

Therefore, probability ofgetting ‘an even number as the sum

number of favorable outcomesP(E10) = Total number of possible result = 18/36 = 1/2

(xi) gaining a primenumber together the sum:

Let E11 = occasion of acquiring a element number as the sum. The occasions of a prime number as the amount will it is in E11 = <(1, 1), (1, 2), (1, 4), (1, 6), (2, 1), (2, 3), (2, 5), (3, 2), (3, 4), (4, 1), (4, 3), (5, 2), (5, 6), (6, 1), (6, 5)> = 15

Therefore, probability ofgetting ‘a element number as the sum’

variety of favorable outcomesP(E11) = Total variety of possible outcome = 15/36 = 5/12

(xii) gaining adoublet of also numbers:

Let E12 = event of acquiring a doublet of also numbers. The occasions of a double of even numbers will certainly be E12 = <(2, 2), (4, 4), (6, 6)> = 3

Therefore, probability ofgetting ‘a doublet of also numbers’

number of favorable outcomesP(E12) = Total number of possible outcome = 3/36 = 1/12

(xiii) gaining amultiple of 2 ~ above one die and a lot of of 3 top top the other die:

Let E13 = occasion of acquiring a lot of of 2 ~ above one die and also a lot of of 3 ~ above the other die. The events of a many of 2 on one die and a multiple of 3 ~ above the other die will be E13 = <(2, 3), (2, 6), (3, 2), (3, 4), (3, 6), (4, 3), (4, 6), (6, 2), (6, 3), (6, 4), (6, 6)> = 11

Therefore, probability ofgetting ‘a multiple of 2 on one die and a many of 3 on the other die’

number of favorable outcomesP(E13) = Total variety of possible outcome = 11/36

4. Twodice room thrown. Uncover (i) the odds in favour of obtaining the sum 5, and (ii) theodds versus getting the sum 6.

Solution:

We know that in a single thrown of two die, the full numberof possible outcomes is (6 × 6) = 36.

Let S it is in the sample space. Then,n(S) = 36.

(i) the odds in favour of obtaining the amount 5:

Let E1 be the occasion of getting the amount 5. Then,E1 = (1, 4), (2, 3), (3, 2), (4, 1) ⇒ P(E1) = 4Therefore, P(E1) = n(E1)/n(S) = 4/36 = 1/9⇒ odds in favour the E1 = P(E1)/<1 – P(E1)> = (1/9)/(1 – 1/9) = 1/8.

(ii) the odds versus getting the sum 6:

Let E2 it is in the event of gaining the sum 6. Then,E2 = (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) ⇒ P(E2) = 5Therefore, P(E2) = n(E2)/n(S) = 5/36 ⇒ odds against E2 = <1 – P(E2)>/P(E2) = (1 – 5/36)/(5/36) = 31/5.

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