1. Check it out! 1 1.2.2: Standard Normal Calculations

2. Richie is studying the probabilities associated with rolling four six-sided dice. Richie listed all 1,296 (6 4 ) possible combinations of dice rolls and recorded their sums. Richie’s results are shown in the histogram and table that follow. Use the information to answer the questions. 2 1.2.2: Standard Normal Calculations

3. Probabilities of Sums of Four Dice 3 1.2.2: Standard Normal Calculations Sum of each roll, x Combinations (number of occurrences of each sum) Probability of each roll, P(x) 410.0008 540.0031 6100.0077 7200.0154 8350.0270 9560.0432 10800.0617 111040.0802 121250.0965 131400.1080 (continued)

4. 4 1.2.2: Standard Normal Calculations Sum of each roll, x Combinations (number of occurrences of each sum) Probability of each roll, P(x) 141460.1127 151400.1080 161250.0965 171040.0802 18800.0617 19560.0432 20350.0270 21200.0154 22100.0077 2340.0031 2410.0008 Total1,2961.000

5. 5 1.2.2: Standard Normal Calculations 1.Estimate the mean of the sums. 2.Why is it more likely to roll a sum of 10 than a sum of 5? 3.What is the probability of rolling a sum greater than 20? 4.What is the probability of rolling a sum between 10 and 15 inclusive? 5.Describe the similarities and differences between the distribution of the sums of four six-sided dice and a normal distribution.

6. 1.Estimate the mean of the sums. The mean is the balancing point of the distribution. Since this distribution is symmetric, the mean is 14—the value at the center of the distribution. 6 1.2.2: Standard Normal Calculations

7. 2.Why is it more likely to roll a sum of 10 than a sum of 5? There are many more combinations of 10 than there are of 5. From the middle column of the table, we can see that the sum of 5 can be obtained only 4 ways. There are 80 ways to roll a sum of 10. 7 1.2.2: Standard Normal Calculations

8. 3.What is the probability of rolling a sum greater than 20? To determine the probability of rolling a sum greater than 20, we can add the probabilities associated with rolls of 21, 22, 23, and 24, shown in the table: 0.0154 + 0.0077 + 0.0031 + 0.0008 = 0.0270 We can also work directly with the combinations for 21, 22, 23, and 24. We can find the probability of rolling a sum greater than 20 by adding up these combinations, then dividing the sum by the total number of combinations. 8 1.2.2: Standard Normal Calculations

9. The number of combinations associated with rolls of 21, 22, 23, and 24 are as follows: 20, 10, 4, and 1. 20 + 10 + 4 + 1 = 35 Divide 35 by the total number of combinations (1,296). 35 ÷ 1296 ≈ 0.0270 The probability of rolling a sum greater than 20 is approximately 0.0270. 9 1.2.2: Standard Normal Calculations

10. 4.What is the probability of rolling a sum between 10 and 15 inclusive? To determine the probability of rolling a sum between 10 and 15 inclusive, we can add the probabilities associated with the rolls of 10, 11, 12, 13, 14, and 15. 0.0617 + 0.0802 + 0.0965 + 0.1080 + 0.1127 + 0.1080 = 0.5671 We can also add up the combinations for 10, 11, 12, 13, 14, and 15, then divide the sum by the total number of combinations. 10 1.2.2: Standard Normal Calculations

11. The number of combinations associated with rolls of 10, 11, 12, 13, 14, and 15 are as follows: 80, 104, 125, 140, 146, and 140. 80 + 104 + 125 + 140 + 146 + 140 = 735 Divide this result by the number of combinations. 735 ÷ 1296 ≈ 0.5671 The probability of rolling a sum between 10 and 15 is approximately 0.5671. 11 1.2.2: Standard Normal Calculations

12. 5.Describe the similarities and differences between the distribution of the sums of four six-sided dice and a normal distribution. The overall shape of the histogram has three important similarities to the normal distribution: The distribution is symmetric. Higher probabilities are associated with values near the mean than away from the mean. There are inflection points on either side of the mean at which the graph changes from concave upward to concave downward (on the left) or concave downward to concave upward (on the right). 12 1.2.2: Standard Normal Calculations

13. There are two key differences: The distribution of the sums of the dice is discrete, while the normal distribution is continuous. We cannot roll a sum of 11.5 or 11.75—we jump from 11 to 12. Also, the distribution of sums begins at 4 and ends at 24, so that the probability of rolling any sum less than 4 or greater than 24 is 0. Even though the theoretical values greater than three standard deviations from the mean in a normal distribution are small, they are still greater than 0. 13 1.2.2: Standard Normal Calculations