4. Two dice are rolled. Find the probability that the sum of the numbers is a) 11 b) not 11 d 2,3, or 4 d) a multiple of 3 e) greater than l f) greater than 3

## Step 1: Calculate Total Outcomes<br />### When two dice are rolled, the total number of possible outcomes is $6 \times 6 = 36$.<br /><br />## Step 2: Calculate Favorable Outcomes for Sum of 11<br />### The combinations that result in a sum of 11 are (5, 6) and (6, 5). Thus, there are 2 favorable outcomes.<br /><br />## Step 3: Calculate Probability of Sum of 11<br />### The probability of the sum being 11 is $\frac{2}{36} = \frac{1}{18}$.<br /><br />## Step 4: Calculate Probability of Sum Not 11<br />### The probability of the sum not being 11 is $1 - \frac{1}{18} = \frac{17}{18}$.<br /><br />## Step 5: Calculate Favorable Outcomes for Sum of 2, 3, or 4<br />### The combinations resulting in a sum of 2 are (1,1). The combinations resulting in a sum of 3 are (1,2), (2,1). The combinations resulting in a sum of 4 are (1,3), (2,2), (3,1). Thus, there are $1 + 2 + 3 = 6$ favorable outcomes.<br /><br />## Step 6: Calculate Probability of Sum of 2, 3, or 4<br />### The probability of the sum being 2, 3, or 4 is $\frac{6}{36} = \frac{1}{6}$.<br /><br />## Step 7: Calculate Favorable Outcomes for Sum as a Multiple of 3<br />### The combinations resulting in a sum that is a multiple of 3 (3, 6, 9, 12) are: (1,2), (2,1), (1,5), (2,4), (3,3), (4,2), (5,1), (3,6), (4,5), (5,4), (6,3), (6,6). Thus, there are 12 favorable outcomes.<br /><br />## Step 8: Calculate Probability of Sum as a Multiple of 3<br />### The probability of the sum being a multiple of 3 is $\frac{12}{36} = \frac{1}{3}$.<br /><br />## Step 9: Calculate Probability of Sum Greater Than 1<br />### Since the minimum sum is 2, all outcomes have a sum greater than 1. Thus, the probability is $\frac{36}{36} = 1$.<br /><br />## Step 10: Calculate Favorable Outcomes for Sum Greater Than 3<br />### The total number of outcomes with a sum less than or equal to 3 are: (1,1), (1,2), (2,1). Thus, there are 3 such outcomes. The number of outcomes with a sum greater than 3 is $36 - 3 = 33$.<br /><br />## Step 11: Calculate Probability of Sum Greater Than 3<br />### The probability of the sum being greater than 3 is $\frac{33}{36} = \frac{11}{12}$.