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We need to deconstruct the probabilities here. Let's assume we're talking about 2 fair 6-sided dice. We are looking for the following:
`P((W = odd and B = even) OR (W=even and B = odd))`
Probabilities like this are easy to deconstruct with a few simple rules (see link below for a more formal treatment of problems like this). Anytime you are looking at the word "or" in a statement of probability like this, you can simply use addition. The above probability becomes:
`P(W=odd and B = even) + P(W = even and B = odd)`
Now, we use another simple rule. We know that the roll on the white and black dice are independent events. Of course, the roll of the white die has absolutely no effect on the roll of the black die (as long as the experiment is conducted properly!). Because we need two independent events to occur to fulfill the condition in both probability statements, the probability of each statement is the product of the probabilities of the independent events. That may have been hard to follow...Here is the equation representation of the above statement using this rule:
`P(W=odd)*P(B=even) + P(W=even)*P(B = odd)`
Now, we just need each of the probabilities listed above. In a fair 6-sided die, we know there are 3 odd numbers and 3 even numbers. Therefore, the probability that a fair die lands on an odd number is 1/2, and the probability that a fair die lands on an even number is 1/2, as well. So, the above overall probability is simplified immensely!
`1/2*1/2 + 1/2*1/2 = 1/2`
So our total probability is 1/2, which makes sense because you only have 4 situations:
1) Both dice are even
2) Both dice are odd
3) White is even, black is odd
4) White is odd, black is even
The probability in the problem considers cases 3 and 4, and considering all four cases have the same probability of occurring, it is pretty clear that our probability will be 1/2.
Hope that helps!
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