Download 1.1 Probability

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On average, children run a mile 90 seconds
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Nine percent of Americans carry no cash,
and half carry $20 or less.
The average teen processes 3,700 texts per
month.
….
The Statisticians Objectives
1.
2.
3.
4.
Ask the right questions
Collect useful data
Summarize the data
Make decisions and generalizations
based on the data
5. Turn the data and decisions into new
knowledge
The Frame
Population
u
Sample
Sampling
u
u
u
u
u
u
u
u
u
u
u
u
u
Describe
Inference
Probability
Probability
1. What is the probability that a flipped
coin comes up heads?
2. What is the probability of a randomly
selected card being a king?
3. What is the chance of rolling a 3 or 4 on
a die?
Random Experiments
Outcomes (minimal results)
Events (A, B, C…)
Sample Space (S)
E1: Flip a coin once
– Outcomes: T or H
– Events:
• A={T}
• B={H}
• C={H or T}
– S = {T,H}
E2: Flip two coins
- Outcomes: (H,H) or (H,T) or
(T,H) or (T,T).
- Events:
- A: {One H, one T} = {(H,T), (T,H)}
- B: {at least one H} = {(H,T),(T,H),
(H,H)}
- S = {(T,T); (T,H);(H,T); (H,H) }
E3: Cast two dice
Outcomes: (1,1) or (1,2) or …
(6,6)
Events:
A = {(3,4)}
B ={The sum is greater than 7}
C=…
S = {(1,1);(1,2); … ; (6,6)}
Probability: classical definition
• All outcomes are equally likely
P(A) =
# outcomes in A
Total # outcomes
You can think of the classical definition
of probability as a proportion
E4. Draw a card
A deck of 52 cards:
S={all possible draws}
# outcomes in S = 52
P(a King) = 4/52 = 1/13
P(a Heart) = 13/52 = ¼
P(king of Hearts)= 1/52
Flip a Coin Three Times
•
Outcomes
HHH HHT HTH HTT
THH THT TTH TTT
1. P(HHH) = 1/8 = 0.125
2. P(Two Heads) = 3/8 = 0.375
3. P(At least 2 Heads) = 4/8 = ½ = 0.5
Roll Two Dice
• Outcomes:
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
P(Sum =2) = 1/36 = 0.0278
P(Sum=9) = 4/36 = 1/9 = 0.111
P(Sum=7) = 6/36 = 1/6 = 0.167
Simple Random Sample
• 1000 people in population,
– 250 prefer red to green
– 300 prefer green to red
– The rest don’t care
• Random person
– P(prefers green to red) = 300/1000 = 30%
– P(don’t care) = 450/100 = 0.45
Probability Properties
• 0 ≤ P(A) ≤ 1
• P(A) = 0 → A is impossible
• P(S) = 1
Probability: a general definition
Size of the Event A
P(A) =
Size of the Sample
Space S
Venn diagrams
A
S
𝐴𝑟𝑒𝑎 𝑜𝑓 𝐴
𝑃 𝐴 =
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑆
Unequal Outcomes
•
•
•
•
Assign a probability to each outcome.
All probabilities ≥ 0.
P(A) = sum P of each outcome in A
All probabilities sum to 1.
– P(S) = 1
• All probabilities ≤ 1.
Choose a Mascot
Oscar Kermit Elmo Grover
P
0.1
0.3
0.2
0.4
•P(Kermit or Elmo) = 0.3 + 0.2 = 0.5
•P(Oscar or Kermit or Elmo) = 0.1 + 0.3 + 0.2 = 0.6
•P(Grover) = 0.4
OR Rule
• If A and B can’t both happen:
P(A OR B) = P(A)+P(B)
• A and B are said to be Mutually
Exclusive
Mutually Exclusive
A
B
S
Mutually Exclusive
AÅB=
A and B cannot both happen
Examples:
A=“Draw a King”
A=“Roll a 3”
B = “Draw a Queen”
B = “Roll an even number”
E2: Flip two coins
- A: {One H, one T}
- P(A) = P[(H,T) OR (T,H)}]
=1/4+1/4= 1/2
- B: {at least one H}
- P(B) = P[(H,T) OR (T,H) OR (H,H)}
= 3/4
Complements
• Complement of A
 All outcomes not in A
c
A
c
• P(A ) = 1 – P(A)
• P(Drawing a card other than an Ace)
=1 – 1/13 = 12/13
E2: Flip two coins
- B: {at least one H}
- P(B) = P[(H,T) OR (T,H) OR (H,H)}
= 3/4
Or we could use the complement:
- BC = {no H}
- P(BC) = P(T,T) = ¼
→ P(B) = 1- P(BC) = 1-1/4=3/4
AND Rule
• If A and B are Independent
P(A AND B) = P(A)P(B)
• Independent
• if A occurs, No affect on if B
occurs
• Examples
H and then H
6 and then 2
A and B
A
AÅB
B
Successive Events
• P(Heads and then Tails) =
• P(Roll
) =
• P(Roll 1 and then an even number) =
Probability Rules
• Not ) 1 - Probability
• OR & Mutually Exclusive ) Add
• AND & Independence ) Multiply
OR Rule
• Mutually Exclusive
–Events contain no common
outcomes
–Intersection is empty
–They can’t both happen
• For mutually exclusive events A,B
P(A or B) = P(A) + P(B)
Mutually Exclusive
• Outcomes are mutually Exclusive
–Cast a die: only one number can
happen
–Flip a coin: only one face shows up
• Mutually Exclusive events are NOT
independent
–If A and B are mutually exclusive they
cannot both happen → P(A AND B) = 0
2. OR Rule
• Roll 2 dice,
P(Sum is 7 or 9) = 1/6 + 1/9 = 5/18
• Flip three coins
P(1 H or 3 H) = 3/8 + 1/8 = ½
• Draw a card
P(K or Q) = 1/13 + 1/13 = 2/13
P(Diamond or Heart) = ¼ + ¼ = 1/2
P(K or Diamond) = ????
General OR Rule
• For any events A, B
P(A or B)
= P(A) + P(B) – P(A and B)
A OR B
A
AÅB
B
S
P(King or Heart)
• P(King) = 4/52 = 1/13
• P(Heart) = 13/52 = ¼
• P(King and Heart) = P(King of Hearts) =
1/52
• P(King or Heart) =
= P(King) + P(Heart) – P(King and Heart)
=4/52 + 13/52 – 1/52 = 16/52 = 4/13
Example
P(A) = 1/3 , P(B)= ¼ and P(A and B)
=1/6
A
Compute P(Ac or B)
B
S
P(Ac or B)?
• Use general OR rule
–P(Ac or B)
= P(Ac) + P(B) – P(Ac and B)
• Note that
– P(Ac)= 1- P(A) = 1-1/3 = 2/3
– P(B) = ¼
Not independent
– P(Ac and B)?
AC
A
S
P(B) = P(Ac and B) + P( A and B)
Since they are mutually exclusive
S
AC and B
A
Then
P(Ac and B) = P(B) – P(A and B)
Finally
– P(Ac)= 1- P(A) = 1-1/3 = 2/3
– P(B) = ¼
– P(Ac and B) = P(B)-P(A and B)
= ¼ - 1/6 = 1/12
• From which
P(Ac or B)= 2/3 + ¼ -1/12 = 5/6