Finding Probabilities of Compound Events

Task

Cecil has two six-sided dice, a red one and a white one.

1. If Cecil throws the two dice, what is the probability that the red die is a 1? What is the probability that the sum of the dice is 7?

2. Are the two events described part (a) independent? Explain.
3. What is the probability that the red die is a 2? What is the probability that the sum of the two dice is 10?
4. Are the two events described in part (c) independent? Explain.

IM Commentary

The goal of this task is to study probabilities of compound events. The events described are at the level of the seventh grade standards. This is a deliberate choice so that students can focus on the meaning of independence in probability. Once these calculations have been made, the teacher may wish to explore some more complex scenarios with more dice or with a different context.

The standard S-CP.A.3 also discusses the mathematical meaning of probabilistic independence: events $A$ and $B$ are independent when

In parts (a) and (b) the independence can be seen in the picture: each of the six rows intersects the diagonal in one box and the two events $A$ (the first die is a 1) and $B$ (the sum of the two dice is 7) each have a probability of $\frac{1}{6}$. In parts (c) and (d) the non-independence of the events is also seen immediately from a geometric point of view: the two events have no intersection even though each of them individually is possible.

Solution

1. In the table below the first number denotes the outcome of the red die and the second number the outcome of the white die so $(5,3)$ means that the red die is a 5 and the white die is a 3.

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

   The cases where the first die is a 1 or where the sum is 7 have been highlighted. From the table we can see that the probability that the red die is a 1 is $\frac{1}{6}$ and the probability that the sum of the two dice is 7 is also $\frac{1}{6}$.

2. These two events are independent. To understand why we need more information, however, than we found in (a). If the red die had been a 2, the probability that the sum of the two is 7 would still be $\frac{1}{6}$, the one successful outcome happening when the white die is a 5. More generally, the probability that the sum is a 7 with any specified value of the red die is the same as the probability that the sum is a 7 with no specified value.

   Conversely, the red die is 1 in exactly 1 out the 6 ways of getting 7. So if we ask for the sum of the two dice to be 7, the probability for the red die being a 1 is still $\frac{1}{6}$, the same as it is with no restriction on the sum of the dice. From this and the previous paragraph, we conclude that the two events described in (a) are independent.

3. In the table below the cases where the red die is a 2 or where the sum of the two dice is 10 have been highlighted.

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

   The probability that the first die is a 2 is $\frac{1}{6}$ while the probability that the sum of the two is 10 is $\frac{1}{12}$.

4. As in part (b) we must investigate further to determine if the events are independent. Notice that the probability that the sum of the two dice is 10, provided that the red die is a 2, is 0. Similarly if the first die is a 1, 2, or 3 the probability that the sum of the two dice is 10 is 0. When the first die is a 4, 5, or 6, however, the probability that the sum is 10 is $\frac{1}{6}$. In none of these cases is the probability equal to the probability that the sum of the dice is 10, namely $\frac{1}{12}$. So the first die being 2 and the sum being 10 are not independent events.