Question

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.

Short answer

Yes, independent.

Step-by-step solution

Step 1. Given Information

Two fair rolls, one with a red face and the other with a green face.

Step 2. Concept Used

Definition condition probability:$P\left(A\right|B)=\frac{P(AandB)}{P\left(B\right)}$

Step 3. Explanation 

$P(sumis7greendieshowsat4)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{1}{6}$

Since the red die must be a $3$ in order for the total to be $7$ (knowing that the green die shows at $4$).

$P(sumis7)=\frac{favorableoutcomes}{possibleoutcomes}=\frac{6}{36}=\frac{1}{6}$

Since the number $7$ can be made in a variety of ways, $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$

If the events "sum is $7$" and "green die shows at $4$" are independent, this probability should be similar. As a result, they are self-sufficient.