1. Grab a penny from the front
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 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)

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. 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

9. 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

10. Examples

11. 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

12. 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

13. 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)

14. 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

15. 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)

16. 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

17. 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

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

19. 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

20. 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

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

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

23. 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

24. Survey

25. 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

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

27. 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

28. 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

29. 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

30. Any questions?