Question

a pair of dice is rolled. find the probability of rolling (a) a sum not more than 6, (b) a sum not less than 5, (c) a sum between 4 and 9 (exclusive). (a) how many possible outcomes are there from rolling two dice? (simplify your answer.)

Explanation:

Step1: Calculate total outcomes

Each die has 6 faces. When rolling two dice, by the multiplication - principle, the total number of possible outcomes is $6\times6 = 36$.

Step2: Find outcomes with sum not more than 6

The possible pairs $(x,y)$ such that $x + y\leq6$ are: $(1,1),(1,2),(1,3),(1,4),(1,5),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(4,1),(4,2),(5,1)$. There are 15 such pairs. So the probability $P(\text{sum}\leq6)=\frac{15}{36}=\frac{5}{12}$.

Step3: Find outcomes with sum not less than 5

The total number of outcomes is 36. The number of outcomes with sum less than 5 is 6: $(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)$. So the number of outcomes with sum not less than 5 is $36 - 6=30$. The probability $P(\text{sum}\geq5)=\frac{30}{36}=\frac{5}{6}$.

Step4: Find outcomes with sum between 4 and 9 (exclusive)

The number of outcomes with sum 4 is 3: $(1,3),(2,2),(3,1)$; the number of outcomes with sum 9 is 4: $(3,6),(4,5),(5,4),(6,3)$. The total number of outcomes with sum between 4 and 9 (exclusive) is $36-(3 + 4)=29$. The probability $P(4\lt\text{sum}\lt9)=\frac{29}{36}$.

Step5: Answer for part (a) of the right - hand side question

The total number of possible outcomes from rolling two dice is 36.

Answer:

(a) $\frac{5}{12}$
(b) $\frac{5}{6}$
(c) $\frac{29}{36}$
(a) 36