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Probability and Propositional Logic Essentials
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Probability and Propositional Logic Essentials
Probability and Propositional Logic Essentials
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1
Question
What does probability theory primarily use to handle uncertain situations?
Page 4
Answer
Discussion of events, categories, and hypotheses in which there is no 100% certainty.
2
Question
What example of a probabilistic hypothesis is provided in the introduction?
Page 4
Answer
How will the weather be tomorrow? based on 'it is sunny 10% and rainy 70% of the time.'
3
Question
How are probabilistic hypotheses usually expressed according to the notes?
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Answer
By propositional logic.
4
Question
What is the definition of propositional logic in the notes?
Page 4
Answer
A branch of logic concerned with propositions and their interrelationship.
5
Question
What is a proposition according to the propositional logic section?
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Answer
A possible state or fact, whether it is true or false.
6
Question
How are propositions typically encoded in propositional logic?
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Answer
With letters, for example, 'it is sunny' can be encoded with the letter S.
7
Question
What does a simple proposition sentence express?
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Answer
One fact about the world.
8
Question
What does a compound proposition sentence contain?
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Answer
Several simple sentences associated with different logical relationships.
9
Question
What symbol is used for negation in propositional logic?
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Answer
$\neg$
10
Question
What does the negation symbol $\neg$ do to a statement?
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Answer
Negates the logical statement or one of the logical symbols.
11
Question
What propositional notation means 'it is not sunny' if S is sunny?
Page 5
Answer
$\neg S$
12
Question
What symbol represents conjunction in propositional logic?
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Answer
$\land$
13
Question
What does the conjunction symbol $\land$ refer to?
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Answer
Two or more propositions occurring together.
14
Question
What does $S \land R$ mean if S is sunny and R is rainy?
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Answer
"It's sunny and raining at the same time."
15
Question
What symbol is used for disjunction in propositional logic?
Page 5
Answer
$\lor$
16
Question
What does the disjunction symbol $\lor$ indicate?
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Answer
The occurrence of one of the propositions.
17
Question
What does $S \lor R$ represent if S is sunny and R is rainy?
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Answer
"The weather is sunny or rainy."
18
Question
How does conjunction differ from disjunction according to the notes?
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Answer
Conjunction $\land$ for propositions occurring together; disjunction $\lor$ for one occurring.
19
Question
What does probability provide a quantitative description of?
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Answer
An assumption, event, condition, or proposition.
20
Question
What numerical range do probability values take according to the notes?
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Answer
Between 0 and 1.
21
Question
What characterizes an impossible event?
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Answer
An event that cannot happen at all with probability = 0.
22
Question
What is the example given for an impossible event?
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Answer
$P(R) = 0$ meaning 'it will never rain'.
23
Question
What defines a sure event in probability?
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Answer
An event whose occurrence is 100% sure with probability = 1.
24
Question
What example illustrates a sure event?
Page 6
Answer
$P(S) = 1$ meaning 'it is always sunny'.
25
Question
How does an impossible event differ from a sure event?
Page 6
Answer
Impossible has P=0 (cannot happen); sure has P=1 (always happens).
26
Question
What is the formula for the probability of event A?
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Answer
$P(A) = \frac{\text{number of ways A occurs}}{\text{sample space}}$
27
Question
What is the sample space in probability calculations?
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Answer
All possible outcomes, denoted as $\Omega$.
28
Question
What is the first step to calculate the probability of an event?
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Answer
Calculate the sample space $\Omega$, which is all possible outcomes.
29
Question
In the coin flipping example, what is the sample space $\Omega$?
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Answer
$\Omega = \{\text{Head}, \text{Tail}\}$.
30
Question
What is the probability of getting heads when flipping a coin?
Page 7
Answer
$P(\text{Head}) = \frac{1}{2} = 0.5$.