Answer: No, they are not independent. The two events are dependent events.
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A = event of rolling a five
B = event of rolling an odd number
P(A) = (number of ways to roll a five)/(number of outcomes total)
P(A) = 1/6
P(B) = (number of ways to roll an odd number)/(number of outcomes total)
P(B) = 3/6
P(B) = 1/2
P(A and B) = (number of ways to roll an odd number AND five at the same time)/(number of outcomes total)
P(A and B) = 1/6
If A and B were independent, then P(A and B) = P(A)*P(B) would be true. Let's see if it is or not
P(A and B) = P(A)*P(B)
1/6 = (1/6)*(1/2)
1/6 = 1/12
The result is false, so P(A and B) = P(A)*P(B) isn't true
This means that A and B are dependent events
If we know B has happened, then that changes the probability of A
If we know A has happened, then B is automatically true (since 5 is odd). I.e, P(B|A) = 1
