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Probability Part 1

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?

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Probability Part 1

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  1. Probability Part 1 Grab a penny from the front

  2. 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?

  3. 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….

  4. What is Probability? • Likelihood • Risk • Chance • Odds

  5. Uses (why do I care?) • Genetics • 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)

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

  7. 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)

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

  9. Examples

  10. 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

  11. 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

  12. 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)

  13. 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

  14. 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)

  15. 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

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

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

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

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

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

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

  22. # 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

  23. Survey

  24. 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

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

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

  27. 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

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

  29. Any questions?

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