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The diagram below shows all the possible totals from adding together the results of rolling two fair dice.

a) What is the probability of rolling a total of 5? Give your answer as a fraction in its simplest form.

b) If you rolled a pair of fair dice 180 times, how many times would you expect to roll a total of 5?

\begin{tabular}{|l|l|l|l|l|l|l|}
\hline \multicolumn{1}{|c|}{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}


Sagot :

Let's analyze the problem step by step.

### Part (a): Probability of Rolling a Total of 5

Firstly, let's consider the total number of possible outcomes when rolling two fair dice. Each die has 6 faces, so:
[tex]\[ \text{Total outcomes} = 6 \times 6 = 36 \][/tex]

Next, let's identify the favorable outcomes that result in a total of 5. By going through the table or pairs:

1. (1, 4)
2. (2, 3)
3. (3, 2)
4. (4, 1)

There are 4 pairs that add up to a total of 5.

The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. So, for rolling a total of 5:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{36} \][/tex]

We simplify this fraction:
[tex]\[ \frac{4}{36} = \frac{1}{9} \][/tex]

So, the probability of rolling a total of 5 is:
[tex]\[ \frac{1}{9} \][/tex]

### Part (b): Expected Number of Times to Roll a Total of 5 in 180 Rolls

To find the expected number of times a total of 5 occurs in 180 rolls, we use the probability calculated above. The expected number of occurrences (E) is given by:
[tex]\[ E = \text{Probability} \times \text{Number of trials} \][/tex]

Here, the probability of rolling a total of 5 is [tex]\( \frac{1}{9} \)[/tex] and the number of trials is 180. Thus:
[tex]\[ E = \frac{1}{9} \times 180 \][/tex]

Calculate the expected number of times:
[tex]\[ E = 20 \][/tex]

So, if you roll a pair of fair dice 180 times, you would expect to roll a total of 5 approximately 20 times.