1. Normal Distribution: Finding Probabilities
Silent
Teacher
Intelligent Practice
Narration
Your Turn
𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 85<𝑋<115 =𝑝
𝑃 𝑋<110 =𝑝
Practice

2. 𝑃 85<𝑋<115 =𝑝 Lower: Upper: 𝜎: 𝜇: 𝑃 𝑋<110 =𝑝
Worked Examples
Your Turn
𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 85<𝑋<115 =𝑝
Lower:
Upper: 𝜎: 𝜇:
𝑃 𝑋<110 =𝑝
𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 92<𝑋<107 =𝑝
Lower:
Upper: 𝜎: 𝜇:
𝑃 𝑋<123 =𝑝

3. 𝑃 95<𝑋<105 =𝑝 𝑃 95<𝑋<105 =𝑝 𝑃 90<𝑋<110 =𝑝
𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝
𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝐹𝑜𝑟 𝑋~𝑁 100,20 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 95<𝑋<105 =𝑝
𝑃 95<𝑋<105 =𝑝
𝑃 90<𝑋<110 =𝑝
𝑃 90<𝑋<110 =𝑝
𝑃 90<𝑋<100 =𝑝
𝑃 50<𝑋<100 =𝑝
𝑃 50<𝑋<100 =𝑝
𝐹𝑜𝑟 𝑋~𝑁 90,20 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 40<𝑋<100 =𝑝
𝑃 95<𝑋<105 =𝑝
𝑃 80<𝑋<140 =𝑝
𝑃 90<𝑋<110 =𝑝
𝑃 100<𝑋<160 =𝑝

4. 𝑃 95<𝑋<105 =𝑝 𝑃 95<𝑋<105 =𝑝 𝑃 90<𝑋<110 =𝑝
𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝
𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝐹𝑜𝑟 𝑋~𝑁 100,20 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 95<𝑋<105 =𝑝
𝑃 95<𝑋<105 =𝑝
𝑝=0.3829
𝑝=0.1974
𝑃 90<𝑋<110 =𝑝
𝑃 90<𝑋<110 =𝑝
𝑝=0.6827
𝑝=0.3829
𝑃 90<𝑋<100 =𝑝
𝑃 50<𝑋<100 =𝑝
𝑝=0.3413
𝑝=0.4937
𝑃 50<𝑋<100 =𝑝
𝐹𝑜𝑟 𝑋~𝑁 90,20 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑝=0.5000
𝑃 40<𝑋<100 =𝑝
𝑃 95<𝑋<105 =𝑝
𝑝=0.5000
𝑝=0.1746
𝑃 80<𝑋<140 =𝑝
𝑃 90<𝑋<110 =𝑝
𝑝=9772
𝑝=0.3413
𝑃 100<𝑋<160 =𝑝
𝑝=0.5000

5. 𝑃 𝑋<105 =𝑝 𝑃 𝑋<105 =𝑝 𝑃 𝑋<110 =𝑝 𝑃 𝑋<90 =𝑝 𝑃 𝑋<150 =𝑝
𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝
𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝐹𝑜𝑟 𝑋~𝑁 100,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 𝑋<105 =𝑝
𝑃 𝑋<105 =𝑝
𝑃 𝑋<110 =𝑝
𝑃 𝑋<90 =𝑝
𝑃 𝑋<150 =𝑝
𝑃 𝑋<75 =𝑝
𝐹𝑜𝑟 𝑋~𝑁 100,100 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝐹𝑜𝑟 𝑋~𝑁 105,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 𝑋<105 =𝑝
𝑃 𝑋<105 =𝑝
𝑃 𝑋<90 =𝑝
𝑃 𝑋<110 =𝑝
𝑃 𝑋≤90 =𝑝
𝑃 𝑋<130 =𝑝

6. 𝑃 𝑋<105 =𝑝 𝑃 𝑋<105 =𝑝 𝑃 𝑋<110 =𝑝 𝑃 𝑋<90 =𝑝 𝑃 𝑋<150 =𝑝
𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝
𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝐹𝑜𝑟 𝑋~𝑁 100,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 𝑋<105 =𝑝
𝑃 𝑋<105 =𝑝
𝑝=0.6915
𝑝=0.8413
𝑃 𝑋<110 =𝑝
𝑃 𝑋<90 =𝑝
𝑝=0.8413
𝑝=0.0227
𝑃 𝑋<150 =𝑝
𝑃 𝑋<75 =𝑝
𝑝=1.0000
𝑝=0.0000
𝐹𝑜𝑟 𝑋~𝑁 100,100 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝐹𝑜𝑟 𝑋~𝑁 105,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 𝑋<105 =𝑝
𝑃 𝑋<105 =𝑝
𝑝=0.6915
𝑝=0.5000
𝑃 𝑋<90 =𝑝
𝑃 𝑋<110 =𝑝
𝑝=0.1587
𝑝=0.8413
𝑃 𝑋≤90 =𝑝
𝑃 𝑋<130 =𝑝
𝑝=0.1587
𝑝=1.0000

7. 𝑃 𝑋>105 =𝑝 𝑃 𝑋>105 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋≥95 =𝑝
𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝
𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝐹𝑜𝑟 𝑋~𝑁 100,35 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 𝑋>105 =𝑝
𝑃 𝑋>105 =𝑝
𝑃 𝑋<95 =𝑝
𝑃 𝑋<95 =𝑝
𝑃 𝑋≥95 =𝑝
𝑃 𝑋>64 =𝑝
𝐹𝑜𝑟 𝑋~𝑁 100,25 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝐹𝑜𝑟 𝑋~𝑁 95,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 𝑋>105 =𝑝
𝑃 𝑋>105 =𝑝
𝑃 𝑋<95 =𝑝
𝑃 𝑋<95 =𝑝
𝑃 𝑋≥80 =𝑝
𝑃 𝑋>100 =𝑝

8. 𝑃 𝑋>105 =𝑝 𝑃 𝑋>105 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋<95 =𝑝 𝑃 𝑋≥95 =𝑝
𝐴𝑙𝑙 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 𝑡𝑜 4.𝑑.𝑝
𝐹𝑜𝑟 𝑋~𝑁 100,10 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝐹𝑜𝑟 𝑋~𝑁 100,35 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 𝑋>105 =𝑝
𝑃 𝑋>105 =𝑝
𝑝=0.3085
𝑝=0.1990
𝑃 𝑋<95 =𝑝
𝑃 𝑋<95 =𝑝
𝑝=0.3085
𝑝=0.1990
𝑃 𝑋≥95 =𝑝
𝑃 𝑋>64 =𝑝
𝑝=0.6915
𝑝=1.0000
𝐹𝑜𝑟 𝑋~𝑁 100,25 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝐹𝑜𝑟 𝑋~𝑁 95,5 2 , 𝑓𝑖𝑛𝑑 𝑝 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡:
𝑃 𝑋>105 =𝑝
𝑃 𝑋>105 =𝑝
𝑝=0.1587
𝑝=0.0227
𝑃 𝑋<95 =𝑝
𝑃 𝑋<95 =𝑝
𝑝=0.1587
𝑝=0.5000
𝑃 𝑋≥81 =𝑝
𝑃 𝑋>100 =𝑝
𝑝=0.9999
𝑝=0.1587