Classical probability

1. Probability
•Quantifying the likelihood that something is going to
happen.
•A number from 0 to 1, inclusive
–0 - Impossible
–1 - Certain, guaranteed
–½ - a “toss up”
•Can be expressed as a fraction (in lowest terms),
decimal, or percent
–Usually starts out as a fraction

2. Probability definition: Event
•An event is one occurrence of the activity whose
probability is being calculated.
–E.g., we are calculating the probability of dice, an event is
one roll of the dice.
•A simple event cannot be broken down into smaller
components
–Rolling one dice is a simple event
•A compound event is made up of several simple events
–The probability of a compound event is usually a function of
the component simple events.
–Rolling two dice is a compound event.

3. Probability definitions: Outcome, sample space
•An outcome is one possible result of the event.
–Rolling a five is one possible outcome of rolling one dice
–Rolling a seven is one possible outcome of rolling two dice
•The sample space is the list of all possible outcomes
–One dice: 1, 2, 3, 4, 5, or 6
–Two dice: See next slide
•The size of the sample space is the total number of possible
outcomes
–One dice: sample space size is 6
–Two dice: sample space size is 36
•A success is an outcome that we want to measure
•A failure is an outcome that we do not want to measure
–Failures = Sample space – successes

4. Two Dice Sample Space
First Die
1 2 3 4 5 6
2
nd
Die
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12

5. Probability Symbols and Calculation
•The letter P denotes a probability.
•Capital letters (A, B, C, etc) represent outcomes
•P(A) denotes the probability of outcome A occurring
•Where a success is when outcome A occurs
Number of possible success
( )
Size of sample space
P A=

6. For example: One Dice
•What is the probability of rolling a five with one
dice?
–Sample space: 1 2 3 4 5 or 6
–Sample space size: 6
–Successful rolls:
–Number of successes:
–P(5) =
•What is the probability of rolling an odd number?
–Successful rolls:
–Number of successes:
–P(Prime) =

7. For example: Two Dice
•What is the probability of rolling a five with one
dice?
–Sample space size: 36
–Successful rolls:
–Number of successes:
–P(5) =
•What is the probability of rolling a prime number?
–Number of successes:
–P(Prime) =

8. Types of Probability
•Classical
–AKA Theoretical or
Empirical
–Events and outcomes in
sample space can be
determined from the ‘rules of
the game’
–E.g., Wheel of fortune
•Geometric
–Sample space is some area, a
successful outcome is hitting
some target
•Experimental
–AKA Relative frequency
–Some activity is observed
–Sample space size is the total
number of events observed
–Success is the subset of events
in which out outcome
occurred
–E.g., basketball toss

9. Classical probability: Coin flip
•Event: coin flip
•Sample space: heads or tails
•Sample space size: 2
•Probability of flipping heads
•Sucesses:
•# of Successes
•P(Heads)

10. Classical Probability: Cards
•Event: drawing one (or more) cards
•Sample space: a deck cards, two colors, each color
has two suits, each suit has 13 ranks deuce to ten,
three face cards, ace
•Sample Space size: 52
•What is the probability of drawing a 10 of spades?
•Successes:
•Number of successes:
P(10♠)

11. Classic Classical Probability: Cards
Successes # of success P
P(Jack)
P(Red)
P(Heart)

12. Your turn
•From a deck of cards
•P(Face card) =
•P(Red ace) =
•P(6 or less) =

13. Classical Probability: Collections
•Sample space: a set of items of different characteristics
–Sample space size. We will know the total and numbers of each
characteristics
•Event: Picking one (or more) items with a specific characteristics
•E.g., A box of balls: 4 red, 2 blue, 2 green, 2 yellow, 1 white and
1 black.
•Sample size:
•P(red)
–Number of successes:
•P(Black or white)
–Number of successes:

14. Your Turn
•If all the tokens we in a
bag and picked at
random:
•P(Square)
•P(2)
•P(3 in a triangle)
1
3
2
1
3
2
1
2
3
1
1
2
1
1
1
2
1
1
1
2
3
3

15. Classic Classical Probability
•Collections with multiple characteristics
•P(North) =
•P(Junior) =
•P(South upperclassman) =
FroshSophJuniorSenior
North400 375 325 350
South350 300 325 275

16. Classic Classical Probability
•Collections with multiple characteristics
•P(North) =
•P(Junior) =
•P(South upperclassman) =
FroshSophJuniorSenior
North400 375 325 350
South350 300 325 275

17. Classical Probability: Spinner
•Event: Spinning the wheel
•Outcome: Spinner stops at a space
•Sample space: individual spaces
•Sample space size: # of spaces
•P(1)
•P(red) =
•P(Prime)
1
3
2
4

18. Do now
•A wheel of fortune has 15 spaces and costs 25 cents
to play. If you win, you get a $3 prize
•Another wheel has 10 spaces and also costs 25 cents.
If you win, you get a prize worth $2.25.
•If you were down to your last 25 cents, which wheel
would you play?
•If you had 10 dollars to spend (25 cents at a time),
which wheel would you play?

19. Identifying the events and sample space
•Sometimes we have to
enumerate the sample
space.
•How many ways are
there to arrange the
genders of three
children?
•Sample space size?

20. Questions, always questions
•What is the probability of
having three girls?
•P(one boy)?
•P(Youngest is a boy)?
•P(At least one boy)?

21. More types of probability
•Geometric probability
•The event is hitting a target on some surface.
Area of the target
( )
Area of the surface
P A=

22. Complimentary events
•If A represents the occurrence of an event, then Ā
represents the event not occurring.
• Ā is the compliment of A
•P(Ā) = 1 – P(A)
heads tails
red = black
male = female
redsox = yankees
=

23. Odds
•Odds against are the ratio P(Ā):P(A), reduced to
lowest terms
•Odds in favor are the reciprocal of the odds against
•What are the odds
–Against drawing a red card
–In favor of drawing an ace
–Against rolling a 5

24. Odds
•Payoff odds against: Net profit : Amount bet
•Example: roulette wheel
•The payoff odds for picking one number are 35:1
–If you bet $1, you win $35, plus your original bet.
–How much do you win if you bet $5?
•What are the actual odds?
–38 spots on the wheel
•Casinos are profitable because the payoff odds are
less than the actual odds