Question 1137820
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I'm assuming the set notation refers to the outcome of rolling the die
A = {1,4,5} = rolling a 1, 4 or 5
B = {2,4,5,6} = rolling a 2, 4, 5, or 6


There are 3 ways to roll an outcome in event A, out of 6 total, so
P(A) = 3/6 = 1/2


There are 4 items in set B, so
P(B) = 4/6 = 2/3


If A and B were independent, then this equation would be true
P(A and B) = P(A)*P(B)
P(A and B) = (1/2)*(2/3)
P(A and B) = 1/3
We'll keep this value in mind


Let 
C = the event in which both A and B happen at the same time
In other words, event C is rolling either a 4 or 5 since these values are found in both set A = {1,<font color=blue>4,5</font>} and set B = {2,<font color=blue>4,5</font>,6}. We can say that set C is the intersection of A and B
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C = {4,5}
We have 2 ways to get a value in set C, out of 6 ways to roll a die
P(C) = 2/6
P(C) = 1/3
we get 1/3 just like with the previous computation


We see that P(A and B) = P(C) = P(A)*P(B) is true, so therefore we have shown A and B are independent events.
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