Chapter 5.3, Problem 75E

To determine

Whether the events “sum is 7” and “green die shows a 4” independent.

Answer to Problem 75E

Yes, events “sum is 7” and “green die shows a 4” are independent.

Explanation of Solution

Given information:

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

Two events:

“sum is 7”

And

“green die shows a 4”

Calculations:

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.

When the number of favorable outcomes is divided by the number of possible outcomes, we get the probability.

  $P\left(\text{S7}|\text{G4}\right)=\frac{\text{Number}\text{of}\text{favorable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{1}{6}\approx 0.1667$

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.

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

We have

  $P\left(\text{S7}|\text{G4}\right)\approx 0.1667$

And

  $P\left(\text{S7}\right)\approx 0.1667$

Both probabilities are identical.

Thus,

They are independent.