IME350 Engineering Data Analysis Lecture 3 Probability Course Packet Slides

Some morbid probabilities… Calculating death rates for various activities is a very valuable pursuit for those engaged in selling life insurance…Check out a few of these probabilities*: ACR 2 *Numbers were obtained from various unverified sources on the web

Let’s use the most current death rate of about – 18/100,000 for driving – 1/100,000 for riding the train – 1/100,000 for flying Now compare to some activities that we Bozemanites are quite likely to engage in ACR 3 Downhill skiing death rate = 0.52/100,000 “hiking” death rate = 2.62/100,000 Climbing (rock and ice) death rate = 6.6/100,000 Rafting death rate = 0.55/100,000 “user days” Kayaking death rate = 2.9/100,000 “user days”

ACR 4 Annapurna descent death rate = 7.3/100 All other 8000 m peak descent death rate = 2.3/100 K2 descent death rate = 11.6/100 Keep in mind…. Odds of dying in Russian Roulette are 16.7/100!!!

Now we are digging into some of the basic concepts of probability: Sample spaces, events, probability axioms and properties, counting techniques, conditional probability and independence. 5 Population Sample Probability Inferential Statistics

Characteristics of probability problems… Probability = the numerical measure of the likelihood of occurrence of an event relative to a set of alternative events. Implicitly there is more than one possible outcome (or the problem would be deterministic) Two things have to be defined – Sample space – Event of interest 6

Sample Space ( S ) The set of all possible outcomes in an experiment Simplest example is a space with two possible outcomes: Coin toss, pass-fail inspection, sex of baby at birth, … 7 Pass Fail

Events An event is a subset of outcomes contained in the sample space – Can be simple (single event) – Or compound (more than one outcome) 8 E1={RYB} E2={YGR} E3={RYB,YGR} Say we are choosing sets of three marbles out of a box… Compound event

9 Review of Set Theory 1. Intersection of events 2. Union of events 3. Mutually Exclusive Events 4. Complementary Event

A more meaningful example 10 RARA RBRB 100 lb 20 ft If the load must be placed at even 2 ft intervals, what is the sample space of all pairs of R A and R B ? There are 10 pairs of reaction forces – solve the equilibrium problem to get a set of general equations.

RARA RBRB 100 lb 20 ft RARA RBRB 100 lb Sample Space Now…what if the load could be 100 lb, 200 lb, or 300 lb, and could be placed anywhere on the beam? In this case, we have a deterministic load value, load location varied discretely. If the load must be placed at even 2 ft intervals, what is the sample space of all pairs of R A and R B ? 11

RARA RBRB 300 lb Sample Space 200 lb 100 lb 200 lb 100 lb Sample space for load placement varied continuously, load varied discretely. RARA RBRB 300 lb Sample Space 100 lb Sample space for load placement varied continuously, load varied continuously between 100 lb and 300 lb. 12

Event subspaces and set operations A={R A >=100lb} B={R B >=100lb} RBRB RARA 300 lb 100 lb A RBRB RARA 300 lb 100 lb B 13

14 RARA 300 lb 100 lb A RBRB B RARA 300 lb 100 lb

Counting Techniques Sometimes the sample space is small enough to list all possibilities, or to visualize pictorially, as we have been doing. More often than not we may have to apply some general counting priniciples. – Product rule – Tree diagrams – Permutations – Combinations 15

Product Rule for k-tuples k-tuples are just ordered sets of k numbers, with each number having n k possible values If we have 2 numbers, then we refer to it as an ordered pair. Product rule says we can get the total number of possible outcomes of a set of ordered k- tuples by simply multiplying their n k possible values together. 16

Product rule examples Think about a contractor example (found in your packet)… – We have ordered pairs (Plumber, Drywall); There are 15 plumbing contractors and 5 drywall contractors – This means that our sample space consists of 15*5 = 75 possible outcomes What if we added a third contractor, with only two options? – Now there are 15*5*2 = 150 possible outcomes 17

Tree diagrams Can construct these to help visualize, may not be practical for large sample spaces. 18 Job 1 – 1 st generation Job 2 – 2 nd generation Job 3 – 3 rd generation Count last generation branches for total number of sample space possibilities.

Permutation So far we have looked at sampling with replacement (an element can appear more than once), where the order of the k-tuple matters What if we sample without replacement and order matters? – The k-tuple is formed by selecting successively from this set without replacement so that an element can appear in at most one of the k positions. – In this case, the number of possible values in the sample space is defined by the equation below, where n is the total number of elements in the set and k is the size of the grouping. – Factorial notation: n!=n(n-1)(n-2)…(2)(1); 0!=1 19

Combination Sampling without replacement when order does not matter. We are given n distinct objects and any unordered subset of size k of the objects is called a combination. 20

Product Rule, Permutation, or Combination??? 1. You are looking on Travelocity for a flight to Tahiti for Spring Break. There are three legs of the flight. Each flight leg has 4 possible airlines to choose from. 21

Product Rule, Permutation, or Combination??? 1.You are looking on Travelocity for a flight to Tahiti for Spring Break. There are three legs of the flight. Each flight leg has 4 possible airlines to choose from. PRODUCT RULE: There are k elements and each can be selected from its own set of n k possible values. 22

Product Rule, Permutation, or Combination??? 2. You are playing poker with your pals and each player receives a hand of five cards out of the 52 card deck. 23

Product Rule, Permutation, or Combination??? 2.You are playing poker with your pals and each player receives a hand of five cards out of the 52 card deck. COMBINATION: Sample without replacement, order doesn’t matter 24

Product Rule, Permutation, or Combination??? 3. You and two of your most athletic friends decide to complete in a relay triathlon. How many different ways are there to order the three of you? 25

Product Rule, Permutation, or Combination??? 3.You and two of your most athletic friends decide to complete in a relay triathlon. How many different ways are there to order the three of you? PERMUTATION : Sample without replacement, order matters 26

Homework 27 Problems 1-5 in Probability Course Packet