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We have two distinct dice. $X$ will be the number of pips showing on the top face of the die after the toss. $Y$ is the same for the second die.

What is the expected value $$Z_2 = greatest\; common\; divisor(X,Y) $$

Can anyone help me how to solve that please? I just know the expected value of $X$, which is $3.5$ and $Y$ is also $3.5$ But I don't know what to do in case of the greatest common divisor.

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    I just know the expected value of X, which is 3.5 and Y is also 3.5 But I don't know what to do in case of the greatest common divisor. – 2017-01-28
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    There are only a few possible values of the greatest common divisor. Find all the ways each one can occur, and from that find the probability of each value. Then use the definition of expected value. – 2017-01-28
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    Can you give me an example for one possiblity, so that I can understand the problem. – 2017-01-28
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    Do you know what the greatest common divisor is? Do you know how to compute the greatest common divisor of $4$ and $6$, for example? If not, you probably need to go and study that first. – 2017-01-28

2 Answers 2

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You can make a table. This is mostly a good idea in cases of two dice. The first column and row are the possible outcomes of the two dice each. For each combination the $gcd$ has been inserted. Please check the entries.

$$ \begin{array}[ht]{|p{2cm}|||p{0.5cm}|p{0.5cm}|p{0.5cm}|p{0.5cm}|p{0.5cm}|p{0.5cm}|p{0.5cm}|p{0.5cm}|p{0.5cm}|} \hline \text{ d1/d2 } 1 & 1 &2 &3 &4 &5 &6 \\ \hline \hline \hline 1 &1 &1 &1 &1 &1 &1 \\ \hline 2& 1 & 2 &1 &2 &1&2 \\ \hline 3& 1 &1 &3 &1 &1&3\\ \hline 4 &1 &2 &1&4&1&2 \\ \hline 5 &1 &1&1&1&5&1 \\ \hline 6&1&2&3&2&1&6 \\ \hline \end{array}$$

Each outcome has a probability of $\frac1{36}$. Now it is not difficult to calculate $E(Z_2)$.

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    Mouse/mice; die/dice. $\nexists$ _dices._ – 2017-01-28
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    @BruceET Now I know that. Thanks for this information and your edit. – 2017-01-28
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Make a $6 \times 6$ table, in which each of the 36 cells shows the GCD. I've filled in the first three rows. (This is before morning coffee for me, so better check!) Can you finish?

X\Y:   1  2  3  4  5  6 
1      1  1  1  1  1  1
2      1  2  1  2  1  2
3      1  1  3  1  1  3
...

Then count to find numerators of fractions of which denominators are 36.

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    Thank you very much. In Europe, it is 7 pm:D good morning. – 2017-01-28