Lecture 1 Notes Flashcards by Janelle Newell

What is an event?

The outcome of a process

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A probability model is _________________________.

visual representation (chart, graph, diagram) of the likelihood of an outcome/event.

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sample space (Ω)

all possible outcomes/elements

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What does the probability measure (P) do?

It assigns a probability to every outcome in the sample space (Ω)

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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

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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

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What is another word for a set (a collection of elements)?

Event

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Containment and Subsets:

If E is a subset of F, ____________________.
E ⊆ F

E is in F.
F contains E.

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If E is a subset of F (E ⊆ F), then _____________________.

P(E) ≤ P(F)

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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)

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Denote the complement of E.

E^c

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Probability of a complement

P(E^c)= 1-P(E)

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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

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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

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Set Operation: Intersection

Denote the intersection between E and F.

E∩F

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P(E∩F)=

P(E)*P(F)
The probability of the intersection = the product of the probabilities.

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What happens with the intersection being commutative

E∩F=F∩E

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When 2 events are disjoint, they _______________________.

have nothing in common.

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The intersection between 2 events is disjoint when _______________ (denote).

E∩F=∅

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What does an empty set mean?

There are no outcomes/intersections between events.

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∅

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empty set symbol

Probability of an outcome in an empty set

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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

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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)

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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

Lecture 1 Notes Flashcards

(50 cards)