Lecture 1 Notes Flashcards
What is an event?
The outcome of a process
A probability model is _________________________.
visual representation (chart, graph, diagram) of the likelihood of an outcome/event.
sample space (Ω)
all possible outcomes/elements
What does the probability measure (P) do?
It assigns a probability to every outcome in the sample space (Ω)
Probability Model 1:
sample space (Ω): a standard deck of cards (52)
E1: the event that a Black card is drawn
E2 the event that a King is drawn
E3 the event that the drawn card has a value of 10 in the game of Blackjack
E1: 26/52=1/2
E2: 4/52=1/13
E3: 16/52=4/13
Probability Model 2:
A fair coin is flipped 3 times.
F1: all three flips are tails
F2: the second flip is heads
F3: at least one flip is heads
(*write out all possible combinations in the sample space (Ω)
TTT, HHH, TTH, HHT, THT, HTH, HTT, THH (Ω=8)
F1: 1/8
F2: 4/8=1/2
F3: 7/8
What is another word for a set (a collection of elements)?
Event
Containment and Subsets:
If E is a subset of F, ____________________.
E ⊆ F
E is in F.
F contains E.
If E is a subset of F (E ⊆ F), then _____________________.
P(E) ≤ P(F)
A complement of an event consists of ___________________________.
a) all outcomes in the sample space (Ω)
b) some outcomes in the sample space (Ω)
c) all outcomes outside the sample space (Ω)
C
Denote the complement of E.
E^c
Probability of a complement
P(E^c)= 1-P(E)
sample space (Ω): a standard deck of cards (52)
complement of E1:
complement of E2:
complement of E3:
(*Remember:)
E1: the event that a Black card is drawn
E2 the event that a King is drawn
E3 the event that the drawn card has a value of 10 in the game of Blackjack
E1^c: the event that a Red card is drawn = 1/2
E2^c: the event that a King is not drawn (any card another than a King is drawn) = 48/52=12/13
E3: the event that the drawn card has a value of less than 10 = 36/52= 9/13
The complement of F1:
The complement of F2:
*Remember
F1: all three flips are tails
F2: the second flip is heads
F3: at least one flip is heads
TTT, HHH, TTH, HHT, THT, HTH, HTT, THH (Ω=8)
F1^c (at least 1 flip is heads) = 7/8
F2^c (the 2nd flip is tails) = 1/2
f3^c (all flips are tails) = 1/8
Hence F1^c = F3
Set Operation: Intersection
Denote the intersection between E and F.
E∩F
P(E∩F)=
P(E)*P(F)
The probability of the intersection = the product of the probabilities.
What happens with the intersection being commutative
E∩F=F∩E
When 2 events are disjoint, they _______________________.
have nothing in common.
The intersection between 2 events is disjoint when _______________ (denote).
E∩F=∅
What does an empty set mean?
There are no outcomes/intersections between events.
∅
empty set symbol
Probability of an outcome in an empty set
P(∅) = 0
sample space (Ω): a standard deck of cards (52)
The intersection E1 ∩ E2
The intersection E1 ∩ E3
The intersection E2 ∩ E3
(*Remember)
E1: the event that a Black card is drawn
E2 the event that a King is drawn
E3 the event that the drawn card has a value of 10 in the game of Blackjack
The intersection E1 ∩ E2 is the event that a Black King is drawn = 2/52 = 1/26
The intersection E1 ∩ E3 is the event that a Black 10,Jack,Queen,King is drawn. = 8/52 = 2/13
The intersection E2 ∩ E3 is the event that a king is drawn. This is because E2 ⊆ E3, so their overlap is just E2 = 4/52 = 1/13
The intersection F1 ∩ F2
The intersection F1 ∩ F3
The intersection F2 ∩ F3
*Remember
F1: all three flips are tails
F2: the second flip is heads
F3: at least one flip is heads
TTT, HHH, TTH, HHT, THT, HTH, HTT, THH (Ω=8)
The intersection F1 ∩ F2 is empty. Indeed F1 and F2 are disjoint
The intersection F1 ∩ F3 is also empty.
The intersection F2 ∩ F3 is again the event F2. Notice that F2 ⊆ F3 and so their overlap is just F2 = 4/8 = 1/2
What does the union between E and F encapsulate?
E, F, and their intersection
Name 2 commutative (order can be reversed) relationships.
union, intersection
Denote the union between E and F.
E∪F or F∪E
Union E1 ∪ E2
Union E1 ∪ E3
Union E2 ∪ E3
(*Remember)
E1: the event that a Black card is drawn
E2 the event that a King is drawn
E3 the event that the drawn card has a value of 10 in the game of Blackjack
The union E1 ∪ E2 is the event that either a Black card, or a King (or
both) is drawn = 28/52 = 7/13
The Union E1 ∪ E3 is the event that either a Black card or a card with a value of 10 in Blackjack (or both) is drawn = 34/52 = 17/26
The union E2 ∪ E3 is just E3. This is again because E2 ⊆ E3 = 8/52 = 2/13
Union of F1 ∪ F2
Union of F1 ∪ F3
Union of F2 ∪ F3
F1: all three flips are tails
F2: the second flip is heads
F3: at least one flip is heads
TTT, HHH, TTH, HHT, THT, HTH, HTT, THH (Ω=8)
The union F1 ∪ F2 is the event that either all flips are tails or the
second flip is heads. = 5/8
The union F1 ∪ F3 is the entire sample space Ω. That is, F1 ∪ F3 covers all possibilities = 1
The union F2 ∪ F3 is equal to F3. = 7/8
Convert unions to intersections (*write formula)
P(E ∪ F) = P(E) + P(F) − P(E ∩ F)
Convert unions to intersections (*write in words)
The probability of the intersection between E and F equals the Probability of E plus the probability of F - the probability of the intersection between E and F (when E and F intersect, the intersection is counted twice, so it must be subtracted once).
If 2 events are disjoint, ____________________________.
there is no intersection, so it becomes an empty set, which we know the probability of is 0 [P(∅) = P(E ∩ F) = 0]
Demorgan’s Laws
(E ∪ F)^c = E^c ∩ F^c (the complement of the union is equal to the intersection of the complements)
(E ∩ F )^c = E^c ∪ F^c (the complement of the intersection is equal to the union of the complements)
Is Set Difference commutative?
No (E \ F does not equal F \ E).
The set difference of E \ F consists of ________________________________.
all outcomes that are in E but not in F (so E excluding the intersection between E and F and F itself).
E \ F
E minus F
The difference E1 \ E2
The difference E2 \ E1
(*Remember)
E1: the event that a Black card is drawn
E2 the event that a King is drawn
E3 the event that the drawn card has a value of 10 in the game of Blackjack
The difference E1 \ E2 is the event that a Black card that is not a King
is drawn = 24/52 = 6/13
The difference E2 \ E1 is the event that the King of Diamonds or the King of Hearts is drawn = 2/52 = 1/26
F2 \ F3
F3 \ F2
F1: all three flips are tails
F2: the second flip is heads
F3: at least one flip is heads
TTT, HHH, TTH, HHT, THT, HTH, HTT, THH (Ω=8)
The event F2 \ F3 is empty. This is because F2 ⊆ F3
The event F3 \ F2 = 3/8
Baye’s rule is used to calculate ______________________.
conditional probability
According to Baye’s rule, P(EIF) means (in words) _____________________.
the probability of E when F happens
According to Baye’s rule, P(EIF) =
[P(FIE) * P(E)]/P(F)
If E and F are independent, the definition of conditional probability can be used.
True
Def. of conditional probability?
P(EIF) = P(E∩F)/P(F)
Total Conditional Probability
P(A)= P(A)= ∑_n*P(E∩F_n)
P = probability
A = any event
B_n = event
symmetric difference
consists of all outcomes in either event, excluding the intersection between both events.
Is symmetric difference commutative (reversible)?
Yes.
Denote the symmetric difference between E and F.
E△F
P(E△F)=
P(E)+P(F)- 2P(E∩F)
E1△E2
E2△E3
(*Remember)
E1: the event that a Black card is drawn
E2 the event that a King is drawn
E3 the event that the drawn card has a value of 10 in the game of Blackjack
E1△E2 is the event that either a non-King Black card or a non-Black
King is drawn
E2△E3 is the event that a 10,J or Q is drawn
F1△F2
F1△F3
F1: all three flips are tails
F2: the second flip is heads
F3: at least one flip is heads
TTT, HHH, TTH, HHT, THT, HTH, HTT, THH (Ω=8)
F1△F2 is equal to F1 ∪ F2 (because they are disjoint) = 5/8
F1△F3 is the entire sample space Ω (it covers all possibilities) = 1
