Dice roll probability, at least 9 total?

Question

If I have two dice with $6$ sides each, what is the probability of me rolling atleast $9$ total? I think I'm correct when thinking that the probability of rolling a $9$ is $\frac{4}{36}$, that is $11.1...\%$, but how do I go from here to calculate the "at least" part?

Answer 1

$\begin{array}{|c|c|c|c|} \hline &\overrightarrow{ D2} & \color{blue}{1} & \color{blue}{2} & \color{blue}{3} & \color{blue}{4} & \color{blue}{5} & \color{blue}{6} \\ \hline D1\downarrow\\\hline \color{blue}{1} &&2 &3 &4 & 5 & 6 & 7\\ \hline \color{blue}{2}&& 3 & 4&5 & 6 & 7 & 8\\ \hline \color{blue}{3} &&4 &5 &6 & 7 & 8 & \color{red}{9}\\ \hline \color{blue}{4} &&5 & 6 & 7 & 8 &\color{red}{9} &\color{red}{10}\\ \hline \color{blue}{5} &&6 &7 &8 &\color{red}{9} &\color{red}{10} &\color{red}{11} \\ \hline \color{blue}{6} && 7 & 8 & \color{red}{9} &\color{red}{10} &\color{red}{11}&\color{red}{12}\\\hline \end{array}$

Every cell containing a number in red, satisfies: $(D1+D2) \ge 9$

Assuming that each die is a fair die, the probability of obtaining any number from 1-6 on each of the two dice is $\frac{1}{6}$.

For example, the probability of obtaining $(D1,D2)=(1,1)$ is $\left(\frac{1}{6}\right)^2 = \left(\frac{1}{36}\right)$

Every individual outcome in the table is obtained with probability $\frac{1}{36}$ as each result is equally likely.

Since there are 36 (6 $\times$ 6 table) total outcomes, the probability will be $\frac{x}{36}$ or an equivalent fraction.

Try to find $x$ via a simple counting method (count how many numbers in red there are).

Answer 2

P(at least a sum of 9) = P(9 <= the sum <= 12) = P((3,6) or (4,5) or (5,4) or (6,3)) + P((4,6) or (5,5) or (6,4)) + P((5,6) or (6,5)) + P((6,6)) = (4 + 3 + 2 + 1)/36 = 5/18

Answer 3

$$\newcommand{\c}[2]{{}^{#1}{\mathbb C}_{#2}} (x+x^2+...+x^6)^2=x^2(x^6-1)^2(x-1)^{-2}=x^2(x^{12}-2x^6+1)\underbrace{(x-1)^{-2}}_{\displaystyle\sum (k+1)x^k}\\ =...-2\underbrace{(2+...+5)}_{14}+...+1\underbrace{(8+...11)}_{38}$$ . $$P=\frac{10}{36}=\frac5{18}$$

Answer 4

A rough and simpleton, but correct, answer: If you have two dices, the total of possibilites are 36, according the Five σ's table. The total of valid results are 10, so the ratio is 10/36 or 5/18.

Answer 5

3+6=9

4+5=9

5+4=9

6+3=9

4+6=10

5+5=10

6+4=10

5+6=11

6+5=11

6+6=12

10 of 36 of the total values are shown here 9-12, so the probability is 5/18 ($0.2$$\overline7$).