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.
