Bayes' Theorem is a method for calculating conditional or posterior probabilities based on prior probabilities and new information. It allows updating the probability of an event occurring after additional evidence is discovered. The document provides an example of using Bayes' Theorem to calculate the probability that a randomly selected defective bolt was manufactured by Machine X, given information about the production rates and defect rates of 3 machines.

1. Bayes’ Theorem

2. Bayes’ Theorem
• A method to compute posterior (conditional) probabilities.

3. Posterior Probability
• A revised probability of an event obtained after getting additional
information.

4. 𝑃 𝐴/𝐵 =
𝑃 𝐵/𝐴 𝑃 𝐴
𝑃 𝐵/𝐴 𝑃 𝐴 + 𝑃 𝐵/ 𝐴 𝑃 𝐴

5. 𝑃 𝑋/𝐴 =
𝑃 𝐴/𝑋 𝑃 𝑋
𝑃 𝐴/𝑋 𝑃 𝑋 + 𝑃 𝐴/𝑌 𝑃 𝑌 + 𝑃 𝐴/𝑍 𝑃 𝑍

6. • In a bolt factory, machines X, Y, and Z manufacture 25%, 35% and 40% of the total
output respectively. Of the total of their output, 5%, 4% and 2%, respectively are
defective. A bolt is drawn at random and is found to be defective. What is the
probability that it was manufactured by machine X?

7. 𝑃 𝑋/𝐴 =
𝑃 𝐴/𝑋 𝑃 𝑋
𝑃 𝐴/𝑋 𝑃 𝑋 + 𝑃 𝐴/𝑌 𝑃 𝑌 + 𝑃 𝐴/𝑍 𝑃 𝑍
𝑃 𝐴 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑔𝑒𝑡𝑡𝑖𝑛𝑔 𝑎 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑙𝑡
𝑃 𝑋 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑎 𝑏𝑜𝑙𝑡 𝑤𝑎𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑋
𝑃 𝑌 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑎 𝑏𝑜𝑙𝑡 𝑤𝑎𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑌
𝑃 𝑍 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑎 𝑏𝑜𝑙𝑡 𝑤𝑎𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑍
𝑃 𝐴/𝑋 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑔𝑖𝑣𝑒𝑛 𝑎 𝑏𝑜𝑙𝑡 𝑖𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑋, 𝑖𝑡 𝑖𝑠 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒
𝑃 𝐴/𝑌 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑔𝑖𝑣𝑒𝑛 𝑎 𝑏𝑜𝑙𝑡 𝑖𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑌, 𝑖𝑡 𝑖𝑠 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒
𝑃 𝐴/𝑍 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑔𝑖𝑣𝑒𝑛 𝑎 𝑏𝑜𝑙𝑡 𝑖𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑍, 𝑖𝑡 𝑖𝑠 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒
𝑃 𝑋/𝐴 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑔𝑖𝑣𝑒𝑛 𝑎 𝑏𝑜𝑙𝑡 𝑖𝑠 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒, 𝑖𝑡 𝑤𝑎𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑋

8. 𝑃 𝑋/𝐴 =
𝑃 𝐴/𝑋 𝑃 𝑋
𝑃 𝐴/𝑋 𝑃 𝑋 + 𝑃 𝐴/𝑌 𝑃 𝑌 + 𝑃 𝐴/𝑍 𝑃 𝑍
𝑃 𝑋 =
25
100
𝑃 𝑌 =
35
100
𝑃 𝑍 =
40
100
𝑃 𝐴/𝑋 =
5
100
𝑃 𝐴/𝑌 =
4
100
𝑃 𝐴/𝑍 =
2
100

9. 𝑃 𝑋/𝐴 =
5
100
×
25
100
5
100
×
25
100
+
4
100
×
35
100
+
2
100
×
40
100
=
1
100
×
1
100
5 × 25
1
100
×
1
100
5 × 25 +
1
100
×
1
100
4 × 35 +
1
100
×
1
100
2 × 40
=
1
100
×
1
100
5 × 25
1
100
×
1
100
{ 5 × 25 + 4 × 35 + 2 × 40 }
=
5 × 25
5 × 25 + 4 × 35 + 2 × 40

10. 𝑃 𝑋/𝐴 =
5 × 25
5 × 25 + 4 × 35 + 2 × 40
=
125
125 + 140 + 80
=
125
345
= 0.362

11. 𝑨𝒍𝒕𝒆𝒓𝒏𝒂𝒕𝒆 𝒎𝒆𝒕𝒉𝒐𝒅:
𝐼𝑓 𝑎 𝑏𝑜𝑙𝑡 𝑖𝑠 𝑑𝑟𝑎𝑤𝑛 𝑎𝑡 𝑟𝑎𝑛𝑑𝑜𝑚 𝑎𝑛𝑑 𝑖𝑠 𝑓𝑜𝑢𝑛𝑑 𝑡𝑜 𝑏𝑒 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒, 𝑎𝑛𝑑 𝑤𝑒 𝑤𝑎𝑛𝑡
𝑡𝑜 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑖𝑡 𝑤𝑎𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 𝑋, 𝑤𝑒 𝑐𝑎𝑛
𝑠𝑖𝑚𝑝𝑙𝑦 𝑑𝑜 𝑠𝑜 𝑏𝑦 𝑓𝑖𝑛𝑑𝑖𝑛𝑔 𝑡ℎ𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑑𝑟𝑎𝑤𝑖𝑛𝑔 𝑎 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑙𝑡
𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑋 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑙𝑜𝑡 𝑜𝑓 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑙𝑡𝑠.
𝐼𝑓 100 𝑏𝑜𝑙𝑡𝑠 𝑎𝑟𝑒 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑖𝑛 𝑎𝑙𝑙, 25 𝑎𝑟𝑒 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑋
𝑂𝑢𝑡 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒, 5% 𝑎𝑟𝑒 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒
𝑆𝑜, 𝑜𝑢𝑡 𝑜𝑓 100, 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑙𝑡𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑋 = 5% 𝑜𝑓 25
=
5
100
× 25 = 1.25
𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑙𝑦, 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑙𝑡𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑌 = 4% 𝑜𝑓 35
=
4
100
× 35 = 1.4
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑙𝑡𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑍 = 2% 𝑜𝑓 40 = 0.8

12. 𝑆𝑜, 𝑜𝑢𝑡 𝑜𝑓 100, 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑙𝑡𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑋 = 5% 𝑜𝑓 25
=
5
100
× 25 = 1.25
𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑙𝑦, 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑙𝑡𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑌 = 4% 𝑜𝑓 35
=
4
100
× 35 = 1.4
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑙𝑡𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑍 = 2% 𝑜𝑓 40 = 0.8
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑖𝑓 𝑎 𝑏𝑜𝑙𝑡 𝑑𝑟𝑎𝑤𝑛 𝑎𝑡 𝑟𝑎𝑛𝑑𝑜𝑚 𝑖𝑠 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒, 𝑖𝑡 𝑤𝑎𝑠
𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑋
=
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑙𝑡𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑜𝑛 𝑋
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑏𝑜𝑙𝑡𝑠 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑
=
1.25
1.25 + 1.4 + 0.8
=
1.25
3.45
= 0.362