Probability question here Thank you in advance I know this i

Probability question here- Thank you in advance!

I know this is very long- can I increase the points in or provide extra compensation in some way?

Two Dice are rolled. A, B and C are events:

A=Sum is eight

B=at least one is five

C= Both are odd.

Find these probabilities.

P(A)

P(B)

P(C)

P(AnB)

P(AnC)

6.P(BnC)

P(AuB)

P(AuC)

P(BuC)

P(A|B)

P(B|A)

P(A|C)

P(C|A)

P(B|C)

Solution

All possible outcomes for rolling two dice is

S = { (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) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) }

n(S) = 36

1. The event A is

A = { (2,6) (3,5) (4,4) (5,3) (6,2) }

n(A) = 5

Therefore, P(A) = n(A)/n(S) = 5/36

2. The event B is

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

n(B) = 11

Therefore, P(B) = n(B)/n(S) = 11/36

3. The event C is

C = { (1,1) (1,3) (1,5) (3,1) (3,3) (3,5) (5,1) (5,3) (5,5) }

n(C) = 9

Therefore, P(C) = n(C)/n(S) = 9/36 = 1/4

4. AnB = { (3,5) (5,3) }

n(AnB) = 2

Therefore, P(AnB) = n(AnB)/n(S) = 2/36 = 1/18

5. AnC = { (3,5) (5,3) }

n(AnC) = 2

Therefore, P(AnC) = n(AnC)/n(S) = 2/36 = 1/18

6. BnC = { (1,5) (3,5) (5,1) (5,3) (5,5) }

n(BnC) = 5

Therefore, P(BnC) = n(BnC)/n(S) = 5/36

7. AuB = { (2,6) (3,5) (4,4) (5,3) (6,2) (1,5) (2,5) (4,5) (5,1) (5,2) (5,4) (5,5) (5,6) (6,5) }

n(AuB) = 14

Therefore, P(AuB) = n(AuB)/n(S) = 14/36 = 7/18

8. AuC = { (2,6) (3,5) (4,4) (5,3) (6,2) (1,1) (1,3) (1,5) (3,1) (3,3) (5,1) (5,5) }

n(AuC) = 12

Therefore, P(AuB) = n(AuB)/n(S) = 12/36 = 1/3

9. BuC = { (2,5) (4,5) (5,2) (5,4) (5,6) (6,5) (1,1) (1,3) (1,5) (3,1) (3,3) (3,5) (5,1) (5,3) (5,5) }

n(BuC) = 15

Therefore, P(AuB) = n(AuB)/n(S) = 15/36 = 5/12

10. P(A|B) = P(AnB)/P(B) = 1/18 x 36/11 = 2/11

11. P(B|A) = P(BnA)/P(A) = 1/18 x 36/5 = 2/5

12. P(A|C) = P(AnC)/P(C) = 1/18 x 4/1 = 2/9

13. P(C|A) = P(CnA)/P(A) = 1/18 x 36/5 = 2/5

14. P(B|C) = P(BnC)/P(C) = 5/36 x 4/1 = 5/9