#---------------------------------------------------------
# File:   MIT18_05S22_in-class3-script.txt
# Author: Jeremy Orloff
#
# MIT OpenCourseWare: https://ocw.mit.edu
# 18.05 Introduction to Probability and Statistics
# Spring 2022
# For information about citing these materials or our Terms of Use, visit:
# https://ocw.mit.edu/terms.
#
#---------------------------------------------------------
Class 3

---------
Jerry
  Slide 1: 

  Slides 2: Announcements/Agenda, (4 minutes)
    ** Note a number of super important concepts today

  Slide 3: Studio 1 recap (2 minutes)
  
  Slide 4: sample space confusion (3 minutes)
    --see slide. This is a very common problem. You ask a student to
      describe the sample space and they give you its size

  Slide 5: conditional probability (2 minutes)
    --**Stress importance** --USED IN ALMOST EVERYTHING WE DO TODAY
    --give red, heart, examples P(H) = 1/4, P(H|Red) = 1/2, P(H | 4)=1/4
    -- Coin flip figure is A = 3 heads, B = first flip is heads

Jen
  Slide 6: Table/concept question -- P(B|A) \ne P(A|B) (5 minutes)
     P(A|B) = 1/8, P(B|A) = 1/5
     POINT: not the same

  Slide 7 Multiplication rule, Total probability (3 minutes)
    --**Important
    --derive from formula for conditional prob.
    --discuss law of total probability.

  Slides 8 Trees and Multiplication Rule (3 minutes)
   -- tree EXAMPLE is from reading
   -- Go through example
      NOTE: multiplying probabilities along a path is the multiplication rule

  Slides 9-12: Concept questions on trees (4 minutes)
    --Do these carefully but try not to spend a lot of time


  Slide 13, 14  Monty Hall Concept question and board question (Work: 8 minutes, discuss 6 minutes)
    Do concept question by show of hands
    Discussion, walk through tree computation.
    (Don't worry about other possible methods yet. We will get to them
     when we do Bayesian updating.)

  Slide 15: Independence (2 minutes)
    --Note symmetry and use mult. rule.
    --Use formula P(A|B) = P(A and B)P(B)/P(A) to see P(A and B) = P(A)P(B)

  Slide 16: Table question (Work 4 minutes, discuss 4 minutes)
    --Answer by show of fingers
    --Emphasize that the thought process is checking if knowing B
    --changes the probability of A, i.e. compare P(A|B) and P(A)
    --Walk through computation on board:
        P(A) = 1/6 P(A|B)= 1/5, P(A|C) = 1/6
        Can discuss in words why B tells you something about the first die but C doesn't

Jerry
  Slide 17 Bayes (4 minutes)
    --****Stress importance
    --Prove by comparing conditional probability formulas

  Slides 18-19 Board question -squirrels (Work 10 minutes, discuss on next slides)

  Slides 20-21 (6 minutes -- may run out of time - do next time)
    Discussion of squirrels
    Prob a random test is correct != prob. a pos test is correct
    --Give algebraic version of Bayes
    --Only discuss table or tree solution if time

  Slide 22 Board question  More Bayes 
    --Only if time (haha), they can work at the board with answer given on Thursday. This is foreshadowing class 11.