Question

If I toss two dice and add up the numbers 100 times, how many times will I get a sum of 8 and/ or two 6’s

Answer 1

Sum of 8 = 13.8888... ≈ 14

Two 6's = 2.7777... ≈ 3

Sum of 8 and two 6's = 0.3858... ≈ 0

Sum of 8 or two 6's = 16.6666... ≈ 17

Explanation:

Create a sample space for the sum of rolling two dice (where x are the die):

`\begin{array}c\cline{1-7} \vphantom{\frac12}x& \textbf{1} &\textbf{2} &\textbf{3} &\textbf{4} &\textbf{5} &\textbf{6}\\\cline{1-7} \textbf{1}&2 &3 & 4& 5&6 &7 \\\cline{1-7} \textbf{2}& 3& 4& 5& 6& 7&8 \\\cline{1-7} \textbf{3}&4 &5 & 6&7 & 8&9 \\\cline{1-7} \textbf{4}&5 &6 &7 & 8&9 &10 \\\cline{1-7} \textbf{5}& 6 &7 & 8&9 &10 & 11\\\cline{1-7} \textbf{6}&7 & 8&9 &10 & 11&12\\\cline{1-7} \end{array}`

Therefore, there are 36 possible outcomes.

Let Event A be getting a sum of 8.

Let Event B be getting two 6's.

From inspection of the the sample space diagram:

`\sf P(A) = (5)/(36)`

`\sf P(B) = (1)/(36)`

Therefore:

`\implies \sf P(A)\;and\;P(B)=(5)/(36) * (1)/(36)=(5)/(1296)`

`\implies \sf P(A)\;or\;P(B)=(5)/(36) +(1)/(36)=(6)/(36)=(1)/(6)`

Given the two dice are tossed 100 times.

Number of times you will get a sum of 8 from 100 tosses of two dice:

`\implies 100 * (5)/(36)=13.8888...`

Number of times you will get a two 6's from 100 tosses of two dice:

`\implies 100 * (1)/(36)=2.7777...`

Number of times you will get a sum of 8 and two 6's from 100 tosses of two dice:

`\implies 100 * (5)/(1296)=0.385802...`

Number of times you will get a sum of 8 or two 6's from 100 tosses of two dice:

`\implies 100 * (1)/(6) = 16.6666...`