.. title: PROB Engineering Probability Class 4 Mon 2022-01-24 .. slug: class04 .. date: 2022-01-24 .. tags: class .. link: .. description: .. type: text .. has_math: true .. sectnum:: .. contents:: Table of contents:: .. Teaching assistants ------------------- We have 2 10-hour grad TAs and an undergrad grader: #. grad: Hao Lu, luh6@ #. grad: Hanjing Wang, wangh36@ #. ugrad: Zehao Li, liz32@ Hao and Hanjing are eager to talk to you and help. Zehao is not allowed to (by RPI's rules); he just grades. We'll assign office hours soon. Alternate emails ---------------- If you would like also to receive emails at another address, tell me. Probability in the real world - enrichment ------------------------------------------ `Statistician Cracks Code For Lottery Tickets `_ Finding these stories is just too easy. Chapter 2 ctd ------------- #. Today: **counting methods**, Leon-Garcia section 2.3, page 41. a. We have an urn with n balls. #. Maybe the balls are all different, maybe not. #. W/o looking, we take k balls out and look at them. #. Maybe we put each ball back after looking at it, maybe not. #. Suppose we took out one white and one green ball. Maybe we care about their order, so that's a different case from green then white, maybe not. #. Applications: a. How many ways can we divide a class of 12 students into 2 groups of 6? #. How many ways can we pick 4 teams of 6 students from a class of 88 students (leaving 64 students behind)? #. We pick 5 cards from a deck. What's the probability that they're all the same suit? #. We're picking teams of 12 students, but now the order matters since they're playing baseball and that's the batting order. #. We have 100 widgets; 10 are bad. We pick 5 widgets. What's the probability that none are bad? Exactly 1? More than 3? #. In the *approval voting* scheme, you mark as many candidates as you please. The candidate with the most votes wins. How many different ways can you mark the ballot? #. In *preferential voting*, you mark as many candidates as you please, but rank them 1,2,3,... How many different ways can you mark the ballot? Leon Garcia, chapter 2, ctd ---------------------------- #. Leon-Garcia 2.3: Counting methods, pp 41-46. a. finite sample space #. each outcome equally probable #. get some useful formulae #. warmup: consider a multiple choice exam where 1st answer has 3 choices, 2nd answer has 5 choices and 3rd answer has 6 choices. i. Q: How many ways can a student answer the exam? #. A: 3x5x6 #. If there are k questions, and the i-th question has :math:`n_i` answers then the number of possible combinations of answers is :math:`n_1n_2 .. n_k` #. 2.3.1 Sampling WITH replacement and WITH ordering a. Consider an urn with n different colored balls. #. Repeat k times: i. Draw a ball. #. Write down its color. #. Put it back. #. Number of distinct ordered k-tuples = :math:`n^k` #. Example 2.1.5. How many distinct ordered pairs for 2 balls from 5? 5*5. #. *Review*. Suppose I want to eat one of the following 4 places, for tonight and again tomorrow, and don't care if I eat at the same place both times: Commons, Sage, Union, Knotty Pine. How many choices to I have where to eat? a. 16 #. 12 #. 8 #. 4 #. something else #. 2.3.2 Sampling WITHOUT replacement and WITH ordering a. Consider an urn with n different colored balls. #. Repeat k times: i. Draw a ball. #. Write down its color. #. Don't put it back. #. Number of distinct ordered k-tuples = n(n-1)(n-2)...(n-k+1) #. *Review*. Suppose I want to visit two of the following four cities: Buffalo, Miami, Boston, New York. I don't want to visit one city twice, and the order matters. How many choices to I have how to visit? a. 16 #. 12 #. 8 #. 4 #. something else #. Example 2.1.6: Draw 2 balls from 5 w/o replacement. a. 5 choices for 1st ball, 4 for 2nd. 20 outcomes. #. Probability that 1st ball is larger? #. List the 20 outcomes. 10 have 1st ball larger. P=1/2. #. Example 2.1.7: Draw 3 balls from 5 with replacement. What's the probability they're all different? a. P = :math:`\small \frac{\text{# cases where they're different}}{\text{# cases where I don't care}}` #. P = :math:`\small \frac{\text{# case w/o replacement}}{\text{# cases w replacement}}` #. P = :math:`\frac{5*4*3}{5*5*5}` #. 2.3.3 Permutations of n distinct objects a. *Distinct* means that you can tell the objects apart. #. This is sampling w/o replacement for k=n #. 1.2.3.4...n = n! #. It grows fast. 1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720, 7!=5040 #. Stirling approx: .. math:: n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n\left(1+\frac{1}{12n}+...\right) #. Therefore if you ignore the last term, the relative error is about 1/(12n). #. Example 2.1.8. # permutations of 3 objects. 6! #. Example 2.1.9. 12 airplane crashes last year. Assume independent, uniform, etc, etc. What's probability of exactly one in each month? a. For each crash, let the outcome be its month. #. Number of events for all 12 crashes = :math:`12^{12}` #. Number of events for 12 crashes in 12 different months = 12! #. Probability = :math:`12!/(12^{12}) = 0.000054` #. **Random does not mean evenly spaced.** #. 2.3.4 Sampling w/o replacement and w/o ordering a. We care what objects we pick but not the order #. E.g., drawing a hand of cards. #. term: **Combinations** of k objects selected from n. Binomial coefficient. .. math:: C^n_k = {n \choose k} = \frac{n!}{k! (n-k)!} #. **Permutations** is when order matters. #. Example 2.20. Select 2 from 5 w/o order. :math:`5\choose 2` #. Example 2.21 # permutations of k black and n-k white balls. This is choosing k from n. #. Example 2.22. 10 of 50 items are bad. What's probability 5 of 10 selected randomly are bad? a. # ways to have 10 bad items in 50 is :math:`50\choose 10` #. # ways to have exactly 5 bad is 3 ways to select 5 good from 40 times # ways to select 5 bad from 10 = :math:`{40\choose5} {10\choose5}` #. Probability is ratio. #. Multinomial coefficient: Partition n items into sets of size :math:`k_1, k_2, ... k_j, \sum k_i=n` .. math:: \frac{n!}{k_1! k_2! ... k_j!} #. 2.3.5. skip **Reading:** 2.4 Conditional probability, page 47- #. New stuff, pp. 47-66: a. **Conditional probability** - If you know that event A has occurred, does that change the probability that event B has occurred? #. **Independence of events** - If *no*, then A and B are independent. #. **Sequential experiments** - Find the probability of a sequence of experiments from the probabilities of the separate steps. #. **Binomial probabilities** - tossing a sequence of *unfair* coins. #. **Multinomial probabilities** - tossing a sequence of *unfair* dice. #. **Geometric probabilities** - toss a coin until you see the 1st head. #. **Sequences of dependent experiments** - What you see in step 1 influences what you do in step 2. Rich Radke's Probability Bites ------------------------------ This is an excellent set of videos for students who want a second viewpoint. https://www.youtube.com/playlist?list=PLuh62Q4Sv7BXkeKW4J_2WQBlYhKs_k-pj Xkcd comic ---------- `Linear Regression `_ Xkcd comic ---------- `Correlation `_