Suppose you roll two fair 6-sided dice, and A is the event that both dice are even, and B is the event that the sum of the dice is 9 or more.
Hint: 2.4, and the very first problem of this worksheet quiz.
(a) Find P(A)
(b) Find P(B)
(c) Find P(A ∪ B)
(d) Find P(Ac ∩ Bc)
Sample space of rolling of 2 fair dice :
a) Event A = Both dice are event = { (2,2) , (2,4) , (2,6) , (4,2) , (4,4) , (4,6) , (6,2) , (6,4) , (6,4) }
P(A) = 9/36 = 1/4
b) Event B = Sum of dice is 9 or more = {(3,6) , (4,5) , (4,6) , (5,4) , (5,5), (5,6), (6,3), (6,4), (6,5), (6,6) }
P(B) = 10/36 = 5/18
c) (A ∩ B) = { (4,6) , (6,4)
, (6,4) }
P(A ∩ B) = 3/36 = 1/12
We know
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
So, P(A ∪ B) = 9/36 + 10/36 - 3/36
= 16/36
= 4/9
d) Now event Ac = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 3), (2, 5), (3, 1), (3, 2), (3, 3), (3, 4),
(3,5), (3, 6), (4, 1), (4, 3), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 3), (6, 5)}
Event Bc = {(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),, (4, 1), (4, 2), (4, 3), (4, 4), (5, 1), (5, 2), (5, 3), (6, 1), (6, 2) }
Ac ∩ Bc = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 3), (2, 5), (3, 1), (3, 2), (3, 3), (3, 4), (3,5), (4, 1), (4, 3), (5, 1), (5, 2), (5, 3), (6, 1)}
So, P(Ac ∩ Bc ) = 20/36 = 5/9
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