Question

You roll a die twice.a. The sum of the outcomes is 5. What is the probability the first roll was a three?b. The first outcome is a 3. What is the probability that the sum is five?

Answer 1

a.)

Let us list down the possible outcomes in which the sum of the outcomes is 5:

$\begin{gathered} \lbrace1,4\rbrace \\ \lbrace2,3\rbrace \\ \lbrace3,2\rbrace \\ \lbrace4,1\rbrace \end{gathered}$

Since there are four possible outcomes in which the sum would be 5:

$\begin{gathered} P=\text{ probability of getting a sum of 5}\times\text{ probability of getting a 3 first roll} \\ P=\frac{1}{9}\times\frac{1}{4}=\frac{1}{36} \end{gathered}$

b.)

Now, we list down the possible outcomes in which we get a 3 on the first roll:

$\begin{gathered} \lbrace3,1\rbrace \\ \lbrace3,2\rbrace \\ \lbrace3,3\rbrace \\ \lbrace3,4\rbrace \\ \lbrace3,5\rbrace \\ \lbrace3,6\rbrace \end{gathered}$

Here, we can see that there are 6 possible outcomes that we get a three on our first roll. Since there are 36 possible outcomes when rolling a die twice:

$\begin{gathered} P=\text{ possibility of rolling a 3}\times\text{ possibility of getting a sum of 5 given that the first roll is 3} \\ P=\frac{6}{36}\times\frac{1}{6} \\ P=\frac{1}{36} \end{gathered}$