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.
