suppose a two sided polygons are randomly tossed. assuming the tetrahedrons are

Question

suppose a two sided polygons are randomly tossed. assuming the tetrahedrons are weighted fair, determine the set of all possible outcomes Z. Assume each face is numbered 1, 2, 3 and 4 . Let the sets A, B, C and D represent the following events: A: the sum of the toss is even. B: the sum of the toss is odd. C: the sum of the toss is a number less than 6. D: the toss yielded the same number on each upturned face. 1- find P(A), P(B), P(C), P(AnB), P(AuB), P(BuC), and P (BnCnD). 2- verify P ((AuB)complement)=P(Acomplement n Bcomplement)

Explanation / Answer

There are two tetrahedron dice with numbers 1, 2, 3 and 4.

There are total 4^2 = 16 possible outcomes

Event A: the sum of toss is even {(1,1), (1, 3), (2, 2), (2, 4), (3, 1), (3, 3), (4,2), (4,4)}

n(A) = 8

P(A) = n(A)/n(S) = 8/16 = 1/2 = 0.5

Event B: the sum of the toss is odd

n(B) = 8

P(B) = n(B)/n(S) = 8/16 = 0.5

Event C: the sum of the toss is a number less than 6, {(1,1), (1,2), (1,3), (1,4), (2,1) (2,2) (2,3) (3,1) (3,2), (4,1)}

n(C) = 10

P(C) = 10/16 = 5/8 = 0.625

Event D: the toss yielded the same number on each upturned face

n(D) = 4

P(D) = 4/16 = 1/4 = 0.25

n(AnB) = 0

P(AnB) = 0

n(AuB) = 16

P(AuB) = 16/16 = 1

n(BuC) = { (1,2), (1,4), (2,1) , (2,3) , (3,2), (4,1)} = 6

P(BuC) = 6/16 = 0.375

n(BnCnD) = 0

P(BnCnD) = 0

2)

P(AuB) = 1, P(AuB)' = 0

A complement is the event with sum equals to odd numbers i.e. event B

B complement is the event with sum equls to even numbers i.e. event A

Hence P(A'nB') = 0

Hence, P ((AuB)complement)=P(Acomplement n Bcomplement) proved

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