Question

you roll two fair dice.
a) if you were to list the sample space for this problem, how many equally likely outcomes would there be?
b) find the probability of getting a sum of 8.
leave your answer as a fraction.
c) find the probability of getting a 4 on the first die.
leave your answer as a fraction.
d) find the probability of getting a sum of 8 and a 4 on the first die.
leave your answer as a fraction.
e) find the probability of getting a sum of 8 or a 4 on the first die.
leave your answer as a fraction.
f) find the probability of getting doubles.
leave your answer as a fraction.

Explanation:

Step1: Calculate sample - space size

Each die has 6 possible outcomes. When rolling two dice, by the multiplication principle, the number of outcomes in the sample space $n(S)=6\times6 = 36$.

Step2: Find outcomes with sum of 8

The pairs of numbers on the two - dice that sum to 8 are $(2,6),(3,5),(4,4),(5,3),(6,2)$. So $n(\text{sum}=8)=5$. The probability $P(\text{sum}=8)=\frac{n(\text{sum}=8)}{n(S)}=\frac{5}{36}$.

Step3: Find outcomes with 4 on first die

The pairs with 4 on the first die are $(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)$. So $n(\text{first}=4)=6$. The probability $P(\text{first}=4)=\frac{n(\text{first}=4)}{n(S)}=\frac{6}{36}=\frac{1}{6}$.

Step4: Find outcomes with sum of 8 and 4 on first die

The only pair that satisfies both is $(4,4)$. So $n(\text{sum}=8\cap\text{first}=4)=1$. The probability $P(\text{sum}=8\cap\text{first}=4)=\frac{n(\text{sum}=8\cap\text{first}=4)}{n(S)}=\frac{1}{36}$.

Step5: Use the addition rule for probability

The addition rule is $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Let $A$ be the event of getting a sum of 8 and $B$ be the event of getting a 4 on the first die. Then $P(A\cup B)=\frac{5}{36}+\frac{6}{36}-\frac{1}{36}=\frac{5 + 6-1}{36}=\frac{10}{36}=\frac{5}{18}$.

Step6: Find outcomes of doubles

The doubles are $(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)$. So $n(\text{doubles}) = 6$. The probability $P(\text{doubles})=\frac{n(\text{doubles})}{n(S)}=\frac{6}{36}=\frac{1}{6}$.

Answer:

a) 36
b) $\frac{5}{36}$
c) $\frac{1}{6}$
d) $\frac{1}{36}$
e) $\frac{5}{18}$
f) $\frac{1}{6}$