# Probabilistic Systems Analysis and Applied Probability

- Institution: Massachusetts Institute of Technology
- Department: Electrical Engineering and Computer Science
- Teacher: Prof. John Tsitsiklis
- Term: Fall 2013
- Level: Undergraduate
- Language: English
- Subjects: Mathematics, Computer science, Engineering
- Source: [MIT OpenCourseWare](https://ocw.mit.edu/courses/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/)
- Licence: [CC BY-NC-SA 4.0](https://creativecommons.org/licenses/by-nc-sa/4.0/) (Non-commercial use only.)
- Syllabus: [at the source](https://ocw.mit.edu/courses/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/pages/syllabus/)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

This course introduces students to the modeling, quantification, and analysis of uncertainty. The tools of probability theory, and of the related field of statistical inference, are the keys for being able to analyze and make sense of data. These tools underlie important advances in many fields, from the basic sciences to engineering and management.

Course Format

This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include:

Lecture Videos by MIT Professor John Tsitsiklis

Lecture Slides and Readings

Recitation Problems and Solutions

Recitation Help Videos by MIT Teaching Assistants

Tutorial Problems and Solutions

Tutorial Help Videos by MIT Teaching Assistants

Problem Sets with Solutions

Exams with Solutions

Related Resource

A complementary resource, Introduction to Probability, is provided by the videos developed for an EdX version of 6.041. These videos cover more or less the same content, in somewhat different order, and in somewhat more detail than the videotaped live lectures.

## Lectures

1. [Lecture 1: Probability Models and Axioms Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/b3601b090c5c), 52 min (Video)
2. [The Probability of the Difference of Two Events](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/308a1a8ba937), 6 min (Video)
3. [Geniuses and Chocolates](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/8d5132933054), 9 min (Video)
4. [Uniform Probabilities on a Square](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/1df69fd87973), 10 min (Video)
5. [Lecture 2: Conditioning and Bayes' Rule Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/5e7ea1f092a4), 52 min (Video)
6. [A Coin Tossing Puzzle](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/935ffa387f6e), 9 min (Video)
7. [Conditional Probability Example](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/9a7f3a8b50a1), 15 min (Video)
8. [The Monty Hall Problem](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/aaa8e0d64c1e), 16 min (Video)
9. [Lecture 3: Independence Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/3f7f680df6c4), 47 min (Video)
10. [A Random Walker](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/c73484a7de3e), 6 min (Video)
11. [Communication over a Noisy Channel](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/c35584c7ab63), 20 min (Video)
12. [Network Reliability](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/09001f229481), 8 min (Video)
13. [A Chess Tournament Problem](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/93e7d1f65537), 19 min (Video)
14. [Lecture 4: Counting Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/40042c4c5fdf), 52 min (Video)
15. [Rooks on a Chessboard](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/753ab922417e), 19 min (Video)
16. [Hypergeometric Probabilities](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/e6c60dacd121), 6 min (Video)
17. [Lecture 5: Discrete Random Variables; Probability Mass Functions; Expectations Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/c61ea8b6822b), 51 min (Video)
18. [Sampling People on Buses](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/a745e53f98f9), 12 min (Video)
19. [PMF of a Function of a Random Variable](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/eae158eeb848), 16 min (Video)
20. [Lecture 6: Discrete Random Variable Examples; Joint PMFs Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/132febdf20e8), 51 min (Video)
21. [Flipping a Coin a Random Number of Times](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/d908efcfb140), 9 min (Video)
22. [Joint Probability Mass Function (PMF) Drill 1](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/703cfb837991), 18 min (Video)
23. [The Coupon Collector Problem](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/654101a3c10b), 8 min (Video)
24. [Lecture 7: Multiple Discrete Random Variables Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/1a6c58b1a33d), 51 min (Video)
25. [Joint Probability Mass Function (PMF) Drill 2](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/3a55cd79965f), 14 min (Video)
26. [Lecture 8: Continuous Random Variables Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/aec03d327f14), 51 min (Video)
27. [Calculating a Cumulative Distribution Function (CDF)](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/8e9dd8149e89), 9 min (Video)
28. [A Mixed Distribution Example](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/56f88538fbf1), 14 min (Video)
29. [Mean and Variance of the Exponential](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/a4077f252c25) (YouTube)
30. [Normal Probability Calculation](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/0491b2195950), 6 min (Video)
31. [Lecture 9: Multiple Continuous Random Variables Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/5895769806f1), 51 min (Video)
32. [Uniform Probabilities on a Triangle](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/ffc56dba924b), 23 min (Video)
33. [Probability that Three Pieces Form a Triangle](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/2f26a67e57d7), 13 min (Video)
34. [The Absent Minded Professor](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/1ba1c6bb0f00), 14 min (Video)
35. [Lecture 10: Continuous Bayes' Rule; Derived Distributions Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/cc99e5c4f488), 49 min (Video)
36. [Inferring a Discrete Random Variable from a Continuous Measurement](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/19ebb20d4850), 19 min (Video)
37. [Inferring a Continuous Random Variable from a Discrete Measurement](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/95dee21bb80a), 12 min (Video)
38. [A Derived Distribution Example](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/343d28ee905e), 10 min (Video)
39. [Ambulance Travel Time](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/739267fc39b7), 7 min (Video)
40. [Lecture 11: Derived Distributions; Convolution; Covariance and Correlation Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/98a2a9a40fab), 52 min (Video)
41. [The Difference of Two Independent Exponential Random Variables](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/51873bc0361b), 7 min (Video)
42. [The Sum of Discrete and Continuous Random Variables](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/8b3c7e06fc73), 6 min (Video)
43. [Lecture 12: Iterated Expectations; Sum of a Random Number of Random Variables Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/3fe294c3259d), 48 min (Video)
44. [The Variance in the Stick Breaking Problem](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/024db29aaf43), 12 min (Video)
45. [Widgets and Crates](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/f312f93e3ef7), 11 min (Video)
46. [Using the Conditional Expectation and Variance](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/d02b71af2e66), 11 min (Video)
47. [A Random Number of Coin Flips](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/999318811b5d), 18 min (Video)
48. [A Coin with Random Bias](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/b9f4b24f668b), 23 min (Video)
49. [Lecture 13 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/7554f2598171), 51 min (Video)
50. [Bernoulli Process Practice](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/7c8fb87c134b), 9 min (Video)
51. [Lecture 14 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/a4bf329dfcb2), 53 min (Video)
52. [Competing Exponentials](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/e3e0622cc107), 8 min (Video)
53. [Lecture 15 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/ede84a865f93), 50 min (Video)
54. [Random Incidence Under Erlang Arrivals](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/1485f05772b7), 10 min (Video)
55. [Lecture 16 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/49f28c74a51d), 53 min (Video)
56. [Setting Up a Markov Chain](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/38a211152ba7), 11 min (Video)
57. [Markov Chain Practice 1](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/790f266d50a3) (YouTube)
58. [Lecture 17 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/11d54abbeba2), 52 min (Video)
59. [Lecture 18 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/19759d507616), 52 min (Video)
60. [Mean First Passage and Recurrence Times](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/5525d0745d61) (YouTube)
61. [Lecture 19 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/6def41f5fc5f), 51 min (Video)
62. [Convergence in Probability and in the Mean Part 1](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/0f06cb252cf8) (YouTube)
63. [Convergence in Probability and in the Mean Part 2](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/612e6bee7fe1) (YouTube)
64. [Convergence in Probability Example](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/64654dd35a7f), 8 min (Video)
65. [Lecture 20 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/3c47f615c975), 52 min (Video)
66. [Probabilty Bounds](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/e0357cad6c58), 11 min (Video)
67. [Using the Central Limit Theorem](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/92d990e5a5d8), 12 min (Video)
68. [Lecture 21 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/afaa6b491c22), 49 min (Video)
69. [Lecture 22 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/381f34fc6237), 53 min (Video)
70. [Inferring a Parameter of Uniform Part 1](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/7ecdf02679d5), 25 min (Video)
71. [Inferring a Parameter of Uniform Part 2](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/69018385692a), 20 min (Video)
72. [An Inference Example](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/ca34fc6af60b), 28 min (Video)
73. [Lecture 23 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/cca22883dd21), 50 min (Video)
74. [Lecture 24 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/33c822dfd70e), 52 min (Video)
75. [Lecture 25 Video](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/e06b48863513), 53 min (Video)
76. [Lecture 1: Probability Models and Axioms1](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/35b15ddd53db), 52 min (Video)
77. [The Probability of the Difference of Two Events](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/5b80d79841b0), 6 min (Video)
78. [Geniuses and Chocolates](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/b4449213f5d6), 9 min (Video)
79. [Uniform Probabilities on a Square](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/b90995af194a), 10 min (Video)
80. [Lecture 2: Conditioning and Bayes' Rule2](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/fe8dba6f4753), 52 min (Video)
81. [A Coin Tossing Puzzle](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/15fa09661c95), 9 min (Video)
82. [Conditional Probability Example](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/6b2853c910cb), 15 min (Video)
83. [The Monty Hall Problem](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/1043d27e5813), 16 min (Video)
84. [Lecture 3: Independence](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/eaa003a1fc14), 47 min (Video)
85. [A Random Walker](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/395c037951b9), 6 min (Video)
86. [Communication over a Noisy Channel](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/8fc0c7254bf5), 20 min (Video)
87. [Network Reliability](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/d173acf8f4a7), 8 min (Video)
88. [A Chess Tournament Problem](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/a5cb37c8e006), 19 min (Video)
89. [Lecture 4: Counting](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/69c7655a3cc8), 52 min (Video)
90. [Rooks on a Chessboard](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/90b23546f4cd), 19 min (Video)
91. [Hypergeometric Probabilities](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/12a736030c88), 6 min (Video)
92. [Lecture 5: Discrete Random Variables Probability Mass Functions Expectations](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/f7a13d2c50e4), 51 min (Video)
93. [Sampling People on Buses](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/52d188520c8e), 12 min (Video)
94. [PMF of a Function of a Random Variable](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/fe27fcf5a722), 16 min (Video)
95. [Lecture 6: Discrete Random Variable Examples Joint PMFs](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/a297be78bb07), 51 min (Video)
96. [Flipping a Coin a Random Number of Times](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/b07891ed6cb8), 9 min (Video)
97. [Joint Probability Mass Function (PMF) Drill 1](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/8a6694a2df52), 18 min (Video)
98. [The Coupon Collector Problem](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/c6c22fbaa9f3), 8 min (Video)
99. [Lecture 7: Multiple Discrete Random Variables](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/e4681fdd6286), 51 min (Video)
100. [Joint Probability Mass Function (PMF) Drill 2](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/4976302fd734), 14 min (Video)
101. [Lecture 8: Continuous Random Variables](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/32ad3dfdef3a), 51 min (Video)
102. [Calculating a Cumulative Distribution Function (CDF)](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/48b9259a54f3), 9 min (Video)
103. [A Mixed Distribution Example](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/48197853aab2), 14 min (Video)
104. [Mean and Variance of the Exponential](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/bd16275052e0) (YouTube)
105. [Normal Probability Calculation](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/74c89ab8c44d), 6 min (Video)
106. [Lecture 9: Multiple continuous Random Variables](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/c41f29a781bb), 51 min (Video)
107. [Uniform Probabilities on a Triangle](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/8eeab1f6945c), 23 min (Video)
108. [Probability that Three Pieces Form a Triangle](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/819be283ece5), 13 min (Video)
109. [The Absent Minded Professor](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/cf93599125ea), 14 min (Video)
110. [Lecture 10: Continuous Bayes' Rule Derived Distributions](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/f8ec626c5f0f), 49 min (Video)
111. [Inferring a Discrete Random Variable from a Continuous Measurement](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/eb0e7cc75895), 19 min (Video)
112. [A Derived Distribution Example](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/1860873d91f2), 10 min (Video)
113. [Ambulance Travel Time](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/589c17a3c9e4), 7 min (Video)
114. [Lecture 11: Derived Distributions Convolution Covariance and Correlation](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/cba94c12d8e0), 52 min (Video)
115. [The Difference of Two Independent Exponential Random Variables](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/fa11b0dfbcb1), 7 min (Video)
116. [The Sum of Discrete and Continuous Random Variables](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/72e0becac81b), 6 min (Video)
117. [Lecture 12: Iterated Expectations Sum of a Random Number of Random variables](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/d27dcdd7f8af), 48 min (Video)
118. [The Variance in the Stick Breaking Problem](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/a6e2b6f791ef), 12 min (Video)
119. [Widgets and Crates](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/4be6bc61f45b), 11 min (Video)
120. [Using the Conditional Expectation and Variance](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/fe09dd210856), 11 min (Video)
121. [A Random Number of Coin Flips](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/1a8e05fa1fb5), 18 min (Video)
122. [A Coin with Random Bias](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/8a8f87f3ca24), 23 min (Video)
123. [Lecture 13: Bernoulli Process](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/e5d2e76d1467), 51 min (Video)
124. [Bernoulli Process Practice](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/35abd52fd101), 9 min (Video)
125. [Lecture 14: Poisson Process - I](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/20293b932c50), 53 min (Video)
126. [Competing Exponentials](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/09c55004c7fc), 8 min (Video)
127. [Lecture 15: Poisson Process - II](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/7dcfd1f5fc0c), 50 min (Video)
128. [Random Incidence Under Erlang Arrivals](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/c8cf3d5816d9), 10 min (Video)
129. [Lecture 16: Markov Chains - I](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/611a85691033), 53 min (Video)
130. [Setting Up a Markov Chain](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/613935c99425), 11 min (Video)
131. [Markov Chain Practice 1](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/dbd1e63b94a1) (YouTube)
132. [Lecture 17: Markov Chains - II](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/40006ab928a0), 52 min (Video)
133. [Lecture 18: Markov Chains - III](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/fba7b5c531bd), 52 min (Video)
134. [Mean First Passage and Recurrence Times](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/d65f20a2fd19) (YouTube)
135. [Lecture 19: Weak Law of Large Numbers](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/785e91d150e1), 51 min (Video)
136. [Convergence in Probability and in the Mean Part 1](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/0d3986adaf50) (YouTube)
137. [Convergence in Probability and in the Mean Part 2](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/f84ecab3e2fa) (YouTube)
138. [Convergence in Probability Example](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/ca050cd70373), 8 min (Video)
139. [Lecture 20: Central Limit Theorem](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/f31338e548bd), 52 min (Video)
140. [Probabilty Bounds](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/766c680ac8b9), 11 min (Video)
141. [Using the Central Limit Theorem](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/7c2bc6fabd79), 12 min (Video)
142. [Lecture 21: Bayesian Statistical Inference - I](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/755673732a65), 49 min (Video)
143. [Lecture 22: Bayesian Statistical Inference - II](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/65b30eb3050e), 53 min (Video)
144. [Inferring a Parameter of Uniform Part 1](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/66031355324f), 25 min (Video)
145. [Inferring a Parameter of Uniform Part 2](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/26f7766f5c24), 20 min (Video)
146. [An Inference Example](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/b38b5bf85dfb), 28 min (Video)
147. [Lecture 23 Classical Statistical Inference - I](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/59e1bb04ba51), 50 min (Video)
148. [Lecture 24: Classical Inference - II](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/379f77571805), 52 min (Video)
149. [Lecture 25: Classical Inference - III Course Review](https://invisiblenetwork.ai/learn/mit/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/ebc6092d9c1a), 53 min (Video)