Quiz 1 Flashcards

1
Q

What is a sample space?

A

set whose elements describe possible outcomes in an experiment

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2
Q

What is an event?

A

an event A occurs if the outcome of an experiment is an element of the set A

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3
Q

When are two events disjoint?

A

disjoint if A∩B=∅.
They do not have any outcome in common.
We may equivalently say that they are mutually exclusive.

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4
Q

When does an event A imply B?

A

an event A implies an event B if A ⊂ B

All the elements of A belong to B

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5
Q

What is De Morgan’s Law?

A

(A∪B)^C=A^C ∩ B^C

(A∩B)^C=A^C ∪ B^C

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6
Q

What is probability?

A

A probability P is an assignment of a real value to each event A in a sample space Ω, such that:

  1. P(A) ∈ [0,1]
  2. P(Ω) = 1
  3. If A and B are disjoint,P(A∪B) =P(A) +P(B)
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7
Q

What is P(A^C)

A

P(A^C) = 1−P(A)

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8
Q

What is P(A∪B)?

A

P(A∪B) =P(A) +P(B)−P(A∩B)

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9
Q

What is the probability of an event A when the probability of each outcome is the same?

A

P(A) =|A| / |Ω|

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10
Q

Counting without replacement

A

N! / (N−P)!

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11
Q

Counting with replacement

A

N^P

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12
Q

Selecting an unordered sample from the set

A

(N choose P) =N! / (N−P)!P!

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13
Q

Conditional Probability of A given B?

A

P(A|B) =P(A∩B) / P(B)
P(A∩B) =P(A|B)P(B)
If two events A and B are mutually exclusive,A∩B=∅ so P(A∩B) = 0, and according to the equality above,P(A|B) = 0 as one would expect.

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14
Q

Bayes Formula

A

P(A|B) =P(B|A)P(A) / P(B)

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15
Q

Law of Total Probability

A

P(A) =N∑i=1 P(A|Bi)P(Bi)

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16
Q

Bayes Thm

A

P(Bi|A) =P(A|Bi)P(Bi) / ∑Nj=1P(A|Bj)P(Bj)

17
Q

Two events are said to be independent if?

A

P(A∩B) =P(A)P(B)
P(A∩B) =P(A)P(B)⇔P(A|B) =P(A)
P(A^C|B) = 1−P(A) =P(A^C)

18
Q

Independence for multiple events?

A
Three events A,B, and C are independent if and only if:
P(A∩B) =P(A)P(B)
P(A∩C) =P(A)P(C)
P(B∩C) =P(B)P(C)
P(A∩B∩C) =P(A)P(B)P(C)