Two good dice are rolled simultaneously.
Let A denoted "the sum show is 8" and B the event "two show same number"
Find p(A), p(B), p(A ᑎ B) and p(A ᑌ B)?
Here is the sample space of all 36 ways the dice can possibly fall.
(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)
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The 5 red ones below denote the event A, where the two
numbers add to 8:
(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)
so the probability of A, p(A), is 5 out of 36 or 5/36, because there
are 5 red ones above out of the 36.
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The 6 red ones below denote the event B, where the two
numbers are the same:
(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)
so the probability of B, p(B), is 6 out of 36 or 6/36, which reduces
to 1/6, because there are 6 red ones above out of the 36.
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The 1 red one below denotes the event A ᑎ B, where the two
numbers add up to 8 AND also are the same number.
(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)
so the probability of A ᑎ B, p(A ᑎ B), is 1 out of 36 or 1/36, because
there is only 1 red one above out of the 36.
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The 10 red ones below denote the event A ᑌ B, where the numbers add to 8
OR are the same OR both.
(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)
so the probability of A ᑌ B, p(A ᑌ B), is 10 out of 36 or 10/36, which
reduces to 5/18, because there are 10 red ones out of the 36 above.
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Notice that the formula:
p(A ᑌ B) = p(A) + p(B) - p(A ᑎ B) holds:
5/18 = 5/36 + 1/6 - 1/36
Get LCD on the right of 36
5/18 = 5/36 + 6/36 - 1/36
5/18 = (5+6-1)/36
5/18 = 10/36
5/18 = 5/18
Edwin
