Download Equally Likely Events

Equally Likely Events
Dice come in all shapes
and sizes
Pick a die
• Imagine I let you pick any die from my
collection and roll it once. If you roll a 1, I
give you a dollar.
• Which die do you pick?
• Why?
Probability
• What’s the probability of getting rolling a 1
on a 6 sided die?
• How do you know?
Multiple Outcomes
• This is a double d20. Each of the 20 sides is
numbered 1-10, twice.
• What is the probability of rolling a 1?
• How do you know?
Calculating Probability
number of successful outcomes
P(event) =
number of possible outcomes
but only when.....
Oddly Shaped Dice
• The probability of rolling a 1 on a d6 is 1/6.
• Why is it harder to believe that the
probability of rolling a 7 on this d7 is 1/7?
Calculating Probability
number of successful outcomes
P(event) =
number of possible outcomes
...when all outcomes are equally likely
Roulette
Urn Problems
Which urn has the best probability of
giving an orange stone?
URN A: 25 orange, 75 green.
URN B: 25 orange, 100 green.
URN C: 12 orange, 60 green.
Urn Problems
Which urn has the best probability of
giving an orange stone?
URN A: 25 orange, 75 green.
25/100=1/4
URN B: 25 orange, 100 green.
25/125=1/5
URN C: 12 orange, 60 green.
12/60=1/6
Counting
Calculating Probability
number of successful outcomes
P(event) =
number of possible outcomes
...when all outcomes are equally likely
How many outcomes
are possible?
• If I roll a d6 and a d4?
Number of ways to roll a d4 *
number of ways to roll a d6 =
4*6 =24
1
2
3
4
5
6
1
1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
2
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
4
4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
2d4
• How many ways are there to roll two
different numbers on 2d4?
Number of ways to roll two different
numbers on 2d4
1
2
3
4
1
1, 1
1, 2
1, 3
1, 4
2
2, 1
2, 2
2, 3
2, 4
3
3, 1
3, 2
3, 3
3, 4
4
4, 1
4, 2
4, 3
4, 4
Number
of green
possibilities
*
Number
of
ways
to
roll
two
different
For each green value, there are
number of remaining purple
on
2d4
threenumbers
possible purple
values
possibilities = 4*3 =12
1
2
3
4
1
1, 1
1, 2
1, 3
1, 4
2
2, 1
2, 2
2, 3
2, 4
3
3, 1
3, 2
3, 3
3, 4
4
4, 1
4, 2
4, 3
4, 4
How many trifectas are possible
in the Belmont Stakes?
• 11 horses ran the 2012 Belmont Stakes
• A trifecta is predicting the first three
finishers in order.
• How many different trifectas was it possible
to bet on?
How many trifectas are possible
in the Belmont Stakes?
• 11 horses could take first place.
• 10 remaining horses could take second
place
• 9 remaining horses could take third place
• 11*10*9 = 990
Notation
11*10*9 =
Permutation:
Falling Factorial:
Factorial:
11P3
113
/(11-3)!
11!
If order does not
matter
•
How many ways are there to draw a two card hand from
a 52 card deck?
•
•
•
First card: 52 ways
Second card 51 ways
But: Drawing K♠, Q♢ is the same as drawing Q♢,
K♠.
•
•
I counted every hand twice!
52*51/2= 1326
Drawing 3 from 13
• How many different ways are there to draw
a three card hand from a deck of 13 cards
(only hearts)?
• First card: 13 ways
• Second card: 12 ways
• Third card: 11 ways.
• How many times did I count each hand?
Getting rid of the extra counts
•
How many different ways are there to order
three cards?
•
•
•
•
•
First slot: 3 ways.
Second slot: 2 ways
Third slot: 1 way
3*2*1=6
I can draw three cards in 6 different orders
Drawing 3 from 13
•
How many different ways are there to draw a three
card hand from a deck of 13 cards (only hearts)?
•
•
•
•
•
First card: 13 ways
Second card: 12 ways
Third card: 11 ways.
How many times did I count each hand? 6
13*12*11/6 = 286
Drawing 3 from 13
•
How many different ways are there to draw a three
card hand from a deck of 13 cards (only hearts)?
•
•
•
•
•
First card: 13 ways
Second card: 12 ways
Third card: 11 ways.
How many times did I count each hand? 6
13*12*11/6 = 286
Once more
•
How many different ways are there to draw a three
card hand from a deck of 13 cards (only hearts)?
•
•
•
Number of ways to draw 3 cards from 13: 13P3
Number of ways to arrange 3 cards: 3P3
(13P3)/(3P3) = 1716/6 = 286
Notation
11*10*9 =
Permutation:
Falling Factorial:
Factorial:
11P3
113
/(11-3)!
11!
Combination:
(13P3)/(3P3) =
Combination:
Factorial:
13C3
/[3!(13-3)!]
13!
Just for fun
• How many different starting games of War
are there?
• Number of ways to split a deck into two
halves: 52C26.
• Number of ways to order 26 cards:
• C * 26! * 26! = 1.58*10 .
52
26
68
26P26