Grab a penny from the front Probability Part 1 Grab a penny from the front

TPS If you flip your penny 6 times, how many heads do you think you will get? What is the probability of getting 2 heads?

Was this result predictable? Are there certain rules? Can we use those rules to predict outcomes? 500 people flip a penny 100 times how many should get 50 heads? Two approaches… 1) Have 500 people flip pennies 100 times Ok, we have 10 people… looks like you are flipping the penny 5000 times… 2) define the rules and make predictions We can predict this without flipping a coin 100 theoretical flips with 500 theoretical people sounds better to me….

What is Probability? Likelihood Risk Chance Odds

Uses (why do I care?) Genetics Wildlife Biology Fisheries Biology Heritability  chance of getting certain alleles (chance baby will have blue hair or blonde eyes) Wildlife Biology Mortality (chance of dying) Longevity (chance of living… not dying over period of years) Habitat Selection (chance living here) Fisheries Biology Chance of capture (chance fish makes a poor life choice) Marking and re-capturing fish (chance it just never learns) Health Care Risk assessment (chance bad thing happens) Treatment options (chance that good thing happens)

How do we use probability? Pop quiz right now on astrophysics: Multiple choice (a,b,c,d,e) True or False?

How do we use probability? Two Steps Define Possible Outcomes. Define Probability/Frequency of each outcome P= 1 = 100% or 13/13, etc. P= 0.5 = 50% or ½ (one out of two events) P= 0.25 = 25% or ¼ (one out of four events)

Test Type Step 1 Possible Outcomes Step 2 Define Probabilities True/False Correct Incorrect 0.50 Multiple Choice (A,B,C,D,E) 0.20 0.80

Graphical Representation Step 1  Possible Outcomes Step 2  Probability of Each Outcome 1/6 1/6 1/6 0.50 Roll 1/6 Flip 0.50 1/6 1/6

Examples

Probability: Step 1 Define all possible “events” or “outcomes” Single coin toss – Heads, Tails Single die toss – 1, 2, 3, 4, 5, 6 Fishing – Capture, No capture Squirrel – Lives or Dies

Probability: Step 2 Define probability of each “event” or “outcome” How? Mechanistic basis (know the rules/mechanism) Coin Die Empirical basis (no hard rules) Fishing Squirrel crossing road

Probability of heads? Rolling a 5? Mechanistic basis Step 1: Two Outcomes – Heads or Tails Step 2: Probability – equal among outcomes Probability = 1/possible outcomes (1/2) Step 1: Six Outcomes – 1,2,3,4,5,6 Step 2: Probability – equal among outcomes Probability = 1/possible outcomes (1/6)

Probability of crossing road? Catching a fish? Empirical basis (no known rule/mechanism) Step 1: Outcomes – Alive or Dead Step 2: Probability – variable, unknown Probability = Run trials (simulations) Step 2: Probability Alive = 2/6 Dead = 4/6

Mechanistic Example Mating Squirrels XX female XY male Probability of a newborn being a male? Four Possible outcomes: XX, XX (Female) XY, XY (Male) Equal probability: 2 in 4 Males, 2 in 4 Females… (p = 0.5)

Rules of Probability Division rule The probability of an event is the number of ways an event can occur, divided by the total number of possible events. # Ways that X can happen Probability # Possible Events

2/4 XY, XY XX, XX, XY, XY # Ways that X can happen Probability of Male # Possible Events XY, XY 2/4 XX, XX, XY, XY

Example of Division Rule 3 Coin tosses. Chance 1 head and 2 tails? # Possible Events? # Ways to get 1 head and 2 tails?

Graphical Representation (# Possible Events) Flip 3 Eight Outcomes H Flip 2 Four Outcomes H T Two Outcomes H H Flip 1 T T H H T T H T T

Graphical Representation Flip 1 (H) Flip 2 (T) Flip 3 (H) H H T H H T T H H T T H T T

# with 1 Head 2 Tails? H H T H H T T H H T T H T T

Example of Division Rule Eight possible events, three with 2 tails HHH HHT HTH HTT THH TTH THT TTT

Probability of 2 Tails and 1 Head # Ways that X can happen Probability of 2 Tails and 1 Head # Possible Events HTT, THT, TTH 3/8 HTT, HTH, HHT, HHH, TTT, THH, THT, TTH

Survey

Multiplication Rule of Probabilities How do we calculate the probability of two independent events occurring? Probability of event A AND event B occurring? A does not affect B We’ve kind of done this already

Multiplication Rule of Probabilities How do we calculate the probability of two independent events occurring? Prob and = Prob * Prob

Graphical Representation 0.5 * 0.5 = 0.25 0.5 H 0.5 T 0.5 * 0.5 = 0.25 0.5 0.5 H 0.5 T 0.5 * 0.5 = 0.25 0.5 T 0.5 * 0.5 = 0.25

Multiplication Rule of Probabilities Unequal probabilities? Two dice rolls Prob 1 each roll? Rolling a 1 is a “success” and anything else is a “failure”. Success?  P(roll =1)=1/6 Failure?  P(roll not =1)=5/6

Graphical Representation Roll 2 1 1/6 * 1/6 = 1/36 1/6 Roll 1 1 1/6 >1 1/6 * 5/6 = 5/36 5/6 1/6 1 5/6 * 1/6 = 5/36 5/6 >1 5/6 >1 5/6 * 5/6 = 25/36

Any questions?