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

You are watching: What is the probability of rolling a sum of 6 with two dice When 2 dice room thrown simultaneously, thus variety of event have the right to be 62 = 36 because each die has 1 to 6 number top top its faces. Climate the feasible outcomes are presented in the listed 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 (2, 1) are different outcomes.

Worked-out troubles involving probability for rolling two dice:

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

(i) A is a straightforward event

(ii) B and 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) since A is composed of a single sample point, the is a simple event.

(ii) since both B and also C contain much more than one sample point, each one of them is a compound event.

(iii) due to the fact that A ∩ B = ∅, A and also B are mutually exclusive.

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

Solution:

When 2 dice space rolled, we have actually n(S) = (6 × 6) = 36.

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

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 are not support exclusive.

(ii) Also, A ∪ B ≠ S.

Therefore, A and B space not exhaustive events.

More instances related come the questions on the probabilities because that throwing two dice.

3. two dice space thrown simultaneously. Discover the probability of:

(i) getting six together a product

(ii) obtaining sum ≤ 3

(iii) getting sum ≤ 10

(iv) gaining a doublet

(v) acquiring a sum of 8

(vi) obtaining sum divisible by 5

(vii) getting sum of atleast 11

(viii) obtaining a lot of of 3 together the sum

(ix) obtaining a total of atleast 10

(x) gaining an also number as the sum

(xi) gaining a element number together the sum

(xii) acquiring a double of even numbers

(xiii) obtaining a lot of of 2 ~ above one die and also a many of 3 on the other die

Solution:

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

(i) obtaining six together a product:

Let E1 = occasion of obtaining six together a product. The number who product is 6 will it is in E1 = <(1, 6), (2, 3), (3, 2), (6, 1)> = 4

Therefore, probability ofgetting ‘six together a product’

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

(ii) obtaining sum ≤ 3:

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

Therefore, probability ofgetting ‘sum ≤ 3’

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

(iii) getting sum ≤ 10:

Let E3 = occasion of acquiring sum ≤ 10. The number whose amount ≤ 10 will 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 result = 33/36 = 11/12(iv)getting a doublet:Let E4 = event of obtaining a doublet. The number i m sorry doublet will certainly be E4 = <(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)> = 6

Therefore, probability ofgetting ‘a doublet’

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

(v)getting a amount of 8:

Let E5 = occasion of obtaining 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 sum divisible by 5:

Let E6 = event of gaining sum divisible by 5. The number whose amount 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 number of possible result = 7/36

(vii)getting amount of atleast 11:

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

Therefore, probability ofgetting ‘sum of atleast 11’

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

(viii) acquiring amultiple that 3 together the sum:

Let E8 = event of gaining a lot of of 3 together the sum. The events of a multiple 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 multiple of 3 as the sum’

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

(ix) obtaining a totalof atleast 10:

Let E9 = event of gaining 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 complete of atleast 10’

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

(x) acquiring an evennumber as the sum:

Let E10 = occasion of gaining an also number together the sum. The occasions of an even number together 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 also number together the sum

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

(xi) getting a primenumber together the sum:

Let E11 = occasion of acquiring a element number together the sum. The events of a element number together the sum will be 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’

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

(xii) obtaining adoublet of even numbers:

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

Therefore, probability ofgetting ‘a doublet of even numbers’

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

(xiii) getting amultiple that 2 top top one die and also a lot of of 3 top top the other die:

Let E13 = occasion of acquiring a many of 2 ~ above one die and a lot of of 3 ~ above the various other die. The events of a many of 2 top top 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 lot of of 2 top top one die and a lot of of 3 ~ above the other die’

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

4. Twodice space thrown. Find (i) the odds in favour of getting the sum 5, and (ii) theodds against getting the sum 6.

Solution:

We understand that in a solitary thrown of 2 die, the full numberof feasible outcomes is (6 × 6) = 36.

Let S be the sample space. Then,n(S) = 36.

(i) the odds in favour of acquiring the sum 5:

Let E1 be the occasion of obtaining the sum 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 of E1 = P(E1)/<1 – P(E1)> = (1/9)/(1 – 1/9) = 1/8.

(ii) the odds versus getting the amount 6:

Let E2 be the occasion of getting the amount 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 versus E2 = <1 – P(E2)>/P(E2) = (1 – 5/36)/(5/36) = 31/5.

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5. Two dice, one blue and one orange, are rolled simultaneously. Uncover the probability that getting