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In my practical assignment, I have this question.

Roll two dice and let:

$A:$ sum of $7$;

$B:$ a double;

Are these two events mutually exclusive?

I can do this part but I am not sure what "a double" means.

What I think it means:

• $(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)$. But as far as I know this is called a "doublet".
• $(1,2),(2,4),(3,6)$ One face is the double of the other.

So someone please tell me what it could be.

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3 Answers

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It is entirely impossible to know exactly what the author meant. But it seems to me like a relatively safe assumption that they meant that the two dice show the same result.

"But as far as I know its called doublet." And I usually call it a pair. Many things have several different names. That's just the way it is.

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As the other answerers have pointed out, a double means that both dice show the same number. "Doublet" or "doublets" is a synonym, see the fifth definition here, or the third definition here. Hence the doubles are (in your notation) $(1,1)$, $(2,2)$, $(3,3)$, $(4,4)$, $(5,5)$, $(6,6)$. You can readily see that these six results have sums $2$, $4$, $6$, $8$, $10$, and $12$, respectively, none of which is 7. Hence the events a sum of 7 and doubles are mutually exclusive.

As an extension of this idea (because I feel that a good answer should give you something else to think about), suppose that you have an $n$-sided die. Let $A$ and $B$ be the events $$ A = (\text{the die sum to 7}) \qquad\text{and}\qquad B = (\text{doubles, i.e. both dice show the same number}). $$ Are these two events mutually exclusive? What if we allow the dice to be numbered differently. For example, instead of numbering the die from 1 to $n$, what if we number them from $-n$ to $n$? Are doubles and sums to 7 still mutually exclusive? What if we don't require the numbers to be sequential? What if we don't require the numbering to be integers, but allow any real number? Are the events doubles and sums to 7 still mutually exclusive?

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Getting the same number on both dice.

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