Two fair dice are tossed, and the up face on each die is recorded. Find the probability of observing each of the following events. Answer in fraction form or decimal approximation with 4 decimal places:
:{ The difference between the numbers is 2 }
:{ A 5 does not appear on either die }
:{ The sum of the numbers is odd }

When the two dice are tossed the sample space 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) }

Step-by-step explanation:

a) Let A be an event that the difference between the numbers is 2.

A: { The difference between the numbers is 2 }

A: { (1,3), (2,4), (3,1), (3,5), (4,2), (4,6), (5,3), (6,4)}

P(A) = 8/36 = 4/18 = 2/9

P(A) = 2/9

b)

Let B be the event that a 5 does not appear on either die.

B: { A 5 does not appear on either die }

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

P(B) = 25/36

c)

Let C be the event that the sum of numbers is odd

C: { The sum of the numbers is odd }

C: { (1,2), (1,4), (1,6), (2,1 ), (2,3 ), (2,5 ), (3,2 ), (3,4 ), (3,6 ), (4,1 ), (4,3 ), (4,5 ), (5,2 ), (5,4 ), (5,6 ), (6,1 ), (6,3 ), (6,5 ) }

P(C) = 18/36 = 9/18 = 1/2

P(C) = 1/2

(sin 42)/31. = sinx)/37

sinx = 37 (sin42)/31

= 37× 0.66/31 = 0.787

x= 51.9

other angle opposite to y = 180-51.9-42

= 180- 93.9 =86.1

so

sin42)/31 = sin 86.1)/y

0.021 = 0.997 /y

y= 0.997/0.021 =47.47