Rolling dice Suppose you roll two fair, six-sided dice—one red and one green. Are the events “sum is 7” and “green die shows a 4” independent? Justify your answer. (See Figure 5.2 on page 314 for the sample space of this chance process.)

two fair, six-sided dice—one red and one green. Are the events “sum is 7” and “green die shows a 4”

Two events are independent, if the probability of occurrence of one event does not affect the probability of occurrence of other event.

Let

S7: sum is $7$

G4: green die shows a $4$

We know that

Green die shows a $4$

That means

For the sum to be $7$, red needs to show a $3$.

In this case, the number of favorable outcomes is $1$and number of possible outcomes is $6$.

The probability is calculated by dividing the number of favourable outcomes by the total number of possible possibilities.

$P(\mathrm{S}7\mid \mathrm{G}4)=\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}=\frac{1}{6}\approx 0.1667$

Now,

The possible combinations to make 7 :

$(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$

In this case, since there are six possible combinations to make $7$.

Thus,

The number of favorable outcomes is $6$and number of possible outcomes is $36$.

$P\left(S7\right)=\frac{6}{36}=\frac{1}{6}\approx 0.1667$

We have

$P(\mathrm{S}7\mid \mathrm{G}4)\approx 0.1667$

And

$P(\mathrm{S}7)\approx 0.1667$

Both probabilities are identical.

Thus,

They are independent.