Question

Rolling dice Suppose you roll two fair, six-sided dice—one red and one green. Are the events “sum is 8” 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.)

Short answer

No, events "sum is $8$ " and "green die shows a $4$ " are not independent.

Step-by-step solution

Step 1:Given information

Two fair, six sided dice are rolled-one red and one green.

Two events:

"sum is 8"

And

"green die shows a 4"

Step 2:Calculation

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

Let

S8: sum is $8$

G4: green die shows a $4$$4$

We know that

Green die shows a$4$

That means

For the sum to be $8$, red needs to show a $4$.

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.

$P(\mathrm{S}8\mid \mathrm{G}4)=\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}=\frac{1}{6}\approx 0.1667$

Now,

The possible combinations to make $8$:

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

In this case, since there are five possible combinations to make 8 .

Thus,

The number of favorable outcomes is $5$and number of possible outcomes is $36$

$P(\mathrm{S}8)=\frac{5}{36}\approx 0.1368$

We have

$P(\mathrm{S}8\mid \mathrm{G}4)\approx 0.1368$

And

$P(\mathrm{S}8)\approx 0.1667$

Both probabilities are not identical.

Thus,

They are not independent.

.