Probability Lecture Notes PDF

2/12/25, 11:54 AM OneNote Probability Friday, January 17, 2025 9:45 AM Central to statistics: How likely is it that the result/pattern we see in our samples represents the true...

2/12/25, 11:54 AM OneNote Probability Friday, January 17, 2025 9:45 AM Central to statistics: How likely is it that the result/pattern we see in our samples represents the true reality of the populations? Doesn’t tell us what is TRUE, just what is more likely (probable) Researcher- how likely does it have to be for you to consider it true enough? ○ How much uncertainty are you comfortable with Probability......of an outcome is the proportion od times an event would occur if we observed the random process an infinate number of times...is a mesure of the liklihoof of an envent occuring in a random. Experiment...is a set of rules we use to help charachterize random processes Probability: statistical notation Value (P) between 0-1 ○ Where P=0 is very unlikely ro occur (no chance) ○ And p=1 (100%) will always occur P(A), where A is the event (specific outsome of interest) PRobability of tail (t) when tossing a coin= P(t); ot tossing a head= P(H) P(t)=0.5 P(H)= 0.5 Law of Large Numbers "As more observation are collectefd, thr proportion of occurrences of a particular outcome converges on its probability" Toss a coin 1 million times vs. 10 times 10 tosses, might get 3H, 7T = P(H)= 3/10=0.3 1 millsion tosses, get more accurate result: 490,000H, 510,000T ○ P(h)=0.49, P(T)=0.51 or basicallt 50:50 Finite probability Space and Calculating P Fintit proabilitt space= all the possible oitcpems ○ Tossing coin has 2 possivle oitcomes (H or T) ○ Rolling a die has 6 Multiple event probabilities: Independent and mutually exclusive events- outcome of 1 event does not impact that of 2nd and both events cannot happen together What is the proability of rolling a 3 or a 4 ○ Add the probabilites P(3)= 1/6=0.167 and P(4)=0.167 So P(3 or 4)= 1/6+1/6=1/3= 0.333 What is the probability of NOT rolling a 1? Subtract (Probabilities sum to 1.0) P(not 1)= 1 -P(1)= 1 -0.167 = 0.833 or 83.3% or 5/6 What is the probability of NOT rolling a 3 or a 4? P(not3or4) = 1 –P(3or4) –1 –0.333 = 0.667 or 66.7% or 4/6 or 2/3101 event, 2 preferred outcomes Add a second die to the mix: 2 dice: What is the probability of rolling 1 & 1 in a single roll together? Multiply the probabilities P(1&1)= 0.167 * 0.167 = 0.028 or 2.8% What is the probability of rolling total of 9 on 2 dice? 4 possible events {4,5; 5,4; 3,6; 6,3} in 36 total possible outcomes (finite probability space {1,1; 1,2; 2,1;2,2;....) P(sum9) = 4/36 = 0.111 These are still independent events –roll on 1 die does not influence the roll on the other die BUT both CAN happen together Gambler’s Fallacy: Monte Carlo Fallacy or “hot hand”: event is likely to happen based on previous outcomes Making a decision(bet) based on previous event(s), even though events are not related Probabilities of events change depending on what happened before You toss a coin 9 times = HHHHHHHHHWhat is the probability of tossing H on the next toss?0.5? https://uvic-my.sharepoint.com/personal/lucybryden_uvic_ca/_layouts/15/Doc.aspx?sourcedoc={9faa0b91-e898-498c-a6e1-c5e15b84c2ef}&action=edit&wd=target… 1/3 2/12/25, 11:54 AM OneNote Recency Bias: Inaccurate judgment due to overvaluinf a recent event Or misconceptions of risk (probability of negitive event) due to popularity in media, etc. ○ E.g., after Shark Week on Discovery Channel, ask people the likelihood of being killed by a shark... ○ Are you at risk of dying by homicide? Terrorist attack? Or heart disease? Empirical Determinations of probability: What happens when we do not know the finite probability space? Empirically Estimated Probability: Missing a priori information about all possible outcomes? Ex: us there a higher probabiliy of finding an archaeological site in highland or lowland area? Dig 10 test pits in highland areas, 10 in lowland areas Find archaeological materials in 7 highland and 4 lowland pits P(highland)= 7/10 = 0.7 P(lowland) = 4/10 = 0.4 Context dependent probabilities. May change if ecological, cultural, etc. Factors change (unlike tossing a coin) Example: finding a parking spot on campus https://uvic-my.sharepoint.com/personal/lucybryden_uvic_ca/_layouts/15/Doc.aspx?sourcedoc={9faa0b91-e898-498c-a6e1-c5e15b84c2ef}&action=edit&wd=target… 2/3 2/12/25, 11:54 AM OneNote But- given enough tirals, anything can happen! Also rember probabilies are not deterministic What if we followed 1 millions poeples parlng between 1-2pm 1000 times someone would find a sport 3 days running What would you bet that 2 people in this room share a birthday video. Whya is this useful? Is the event (outcome) plausible by random chance, or might other factors underlie event? ○ E.g., A skeletal sample from a site contains a a high number of child skeletons. Is this expected? Or were children preferentially buried in this location? How do we differentiate likely from unlikely? Staticttically use P