a) What is the probability of rolling a total of 4 with a pair of fair dice? Give your answer in its simplest form.

b) If you rolled a pair of fair dice 360 times, how many times would you expect to roll a total of 4?

[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
+ & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
5 & 6 & 7 & 8 & 9 & 10 & 11 \\
\hline
6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\end{tabular}
\][/tex]



Answer :

a) To find the probability of rolling a total of 4 with a pair of fair dice, we first note all possible outcomes. Each die has 6 faces, so with two dice, there are 6 * 6 = 36 possible outcomes.

Next, we identify the outcomes that sum to 4. They are:
1. (1, 3)
2. (2, 2)
3. (3, 1)

Thus, there are 3 outcomes that result in a total of 4. The probability of rolling a total of 4 with two dice is the number of favorable outcomes divided by the total number of possible outcomes. In fraction form, this probability is:

[tex]\[ \frac{3}{36} = \frac{1}{12} \][/tex]

So, the probability of rolling a total of 4 is [tex]\( \frac{1}{12} \)[/tex].

b) To determine how many times you would expect to roll a total of 4 in 360 rolls of the dice, we use the probability found in part (a). The expected number of times to roll a 4 is given by:

[tex]\[ \text{Expected number of times} = \text{Probability} \times \text{Number of rolls} \][/tex]

Substituting the values we have:

[tex]\[ \text{Expected number of times} = \left( \frac{1}{12} \right) \times 360 = 30 \][/tex]

Therefore, if you roll a pair of fair dice 360 times, you would expect to roll a total of 4 approximately 30 times.