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Questions and Answers
Classical probability may be categorized under the headings of classical and the relative frequency, or a ______, concept of probability.
Classical probability may be categorized under the headings of classical and the relative frequency, or a ______, concept of probability.
posteriori
The probability that the event A will occur is denoted by ______(A).
The probability that the event A will occur is denoted by ______(A).
P
If a random experiment can result in N mutually exclusive and equally likely outcomes, the probability of an event A is defined as the ratio n(A)/______.
If a random experiment can result in N mutually exclusive and equally likely outcomes, the probability of an event A is defined as the ratio n(A)/______.
N
In relative frequency, if some process is repeated N times and an event with characteristic A occurs n times, the relative frequency is ______/N.
In relative frequency, if some process is repeated N times and an event with characteristic A occurs n times, the relative frequency is ______/N.
When throwing a fair die, there are ______ equally likely outcomes.
When throwing a fair die, there are ______ equally likely outcomes.
In the sample space for tossing a coin three times, there are a total of ______ possible sequences.
In the sample space for tossing a coin three times, there are a total of ______ possible sequences.
Let A be the event that two or more heads appear ______, and B that all the tosses are the same.
Let A be the event that two or more heads appear ______, and B that all the tosses are the same.
Since the outcomes are equally likely and mutually exclusive, the probability of each outcome in the coin toss is ______.
Since the outcomes are equally likely and mutually exclusive, the probability of each outcome in the coin toss is ______.
The possible outcomes when tossing a coin are heads (denoted by H) and ______ (denoted by T).
The possible outcomes when tossing a coin are heads (denoted by H) and ______ (denoted by T).
The set of all possible outcomes of an experiment is called the ______ Space.
The set of all possible outcomes of an experiment is called the ______ Space.
When two coins are tossed, the sample space can be represented as S = { HH , HT , ______ , TT }.
When two coins are tossed, the sample space can be represented as S = { HH , HT , ______ , TT }.
The outcomes of two tossed coins can also be represented as ordered pairs of 1 and ______.
The outcomes of two tossed coins can also be represented as ordered pairs of 1 and ______.
When a coin is tossed three times, the sample space consists of ______ outcomes.
When a coin is tossed three times, the sample space consists of ______ outcomes.
If a coin is tossed repeatedly until a ______ occurs, the sample space is given as S = {H, TH, TTH...}.
If a coin is tossed repeatedly until a ______ occurs, the sample space is given as S = {H, TH, TTH...}.
A possible sample space for the number of tosses required to obtain a head would be the set of all positive ______.
A possible sample space for the number of tosses required to obtain a head would be the set of all positive ______.
In a scenario where a light bulb is measured until it burns out, the ______ of operation is assessed.
In a scenario where a light bulb is measured until it burns out, the ______ of operation is assessed.
If D1 is the event that the first fuse is defective, then P ( D1 ) = ____
If D1 is the event that the first fuse is defective, then P ( D1 ) = ____
The events B1, B2,..., Bk constitute a _______ of the sample space S.
The events B1, B2,..., Bk constitute a _______ of the sample space S.
To find the probability of occurrence of an event A that can occur with one of the B's events, the rule of _____ probability is required.
To find the probability of occurrence of an event A that can occur with one of the B's events, the rule of _____ probability is required.
If P(Bi) > 0 for i = 1, 2,..., k, then for any event A in S, P ( A ) = ____ P ( Bi ).P ( A/ Bi ).
If P(Bi) > 0 for i = 1, 2,..., k, then for any event A in S, P ( A ) = ____ P ( Bi ).P ( A/ Bi ).
The formula used for the probability of the intersection of events D1, D2, and D3 is P( D1 D2 D3 ) = ____.
The formula used for the probability of the intersection of events D1, D2, and D3 is P( D1 D2 D3 ) = ____.
Since the events B1, B2,..., Bk are exhaustive, their union equals ____.
Since the events B1, B2,..., Bk are exhaustive, their union equals ____.
A = A ∩ S = (A ∩ B1) ∪ (A ∩ B2) ∪...∪ (A ∩ Bk) is a representation of ____ events.
A = A ∩ S = (A ∩ B1) ∪ (A ∩ B2) ∪...∪ (A ∩ Bk) is a representation of ____ events.
A ∩ B1, A ∩ B2, ..., A ∩ Bk are also mutually ____.
A ∩ B1, A ∩ B2, ..., A ∩ Bk are also mutually ____.
The sample space for the experiment of tossing a fair die twice is denoted by ______.
The sample space for the experiment of tossing a fair die twice is denoted by ______.
The probability of event A is denoted by P(A) and follows the axioms of probability, starting with Axiom I: P(A) ≥ ______.
The probability of event A is denoted by P(A) and follows the axioms of probability, starting with Axiom I: P(A) ≥ ______.
The events A1 and A2 in the example correspond to { ______ number greater than 5 } and { an odd number less than 9 }, respectively.
The events A1 and A2 in the example correspond to { ______ number greater than 5 } and { an odd number less than 9 }, respectively.
In the example, event A consists of the outcomes {(3, 6), (4, 5), (5, 4), (6, ______)}.
In the example, event A consists of the outcomes {(3, 6), (4, 5), (5, 4), (6, ______)}.
For events A1 and A2, the total number of possible outcomes in the sample space is denoted by ______.
For events A1 and A2, the total number of possible outcomes in the sample space is denoted by ______.
If n(A) = 3, then P(A) can be calculated as P(A) = n(A) / ______.
If n(A) = 3, then P(A) can be calculated as P(A) = n(A) / ______.
There are a total of ______ outcomes when tossing a fair die twice.
There are a total of ______ outcomes when tossing a fair die twice.
According to Axiom II: P(S) must equal ______.
According to Axiom II: P(S) must equal ______.
The eight possible outcomes from flipping a fair coin are HHH, HHT, HTH, THH, HTT, THT, TTH, and ______.
The eight possible outcomes from flipping a fair coin are HHH, HHT, HTH, THH, HTT, THT, TTH, and ______.
Event A includes the outcomes {HHH, ______}
Event A includes the outcomes {HHH, ______}
The intersection of events A and B is ______.
The intersection of events A and B is ______.
The probability of event A is P(A) = ______.
The probability of event A is P(A) = ______.
Theorem 2.10 states that if events A and B are independent, then A and ______ are also independent.
Theorem 2.10 states that if events A and B are independent, then A and ______ are also independent.
To define independence for multiple events, we consider the probability of the intersection of any ______ of these events.
To define independence for multiple events, we consider the probability of the intersection of any ______ of these events.
For three events A, B, and C to be independent, the equation P(A ∩ B ∩ C) must equal P(A) · P(B) · ______.
For three events A, B, and C to be independent, the equation P(A ∩ B ∩ C) must equal P(A) · P(B) · ______.
Events can be pairwise independent without being completely ______.
Events can be pairwise independent without being completely ______.
Two events A and B are said to be ______ if the occurrence or nonoccurrence of either one does not affect the probability of the occurrence of the other.
Two events A and B are said to be ______ if the occurrence or nonoccurrence of either one does not affect the probability of the occurrence of the other.
If two events A and B are independent, then P(B / A) = P(______).
If two events A and B are independent, then P(B / A) = P(______).
The formal definition of independence states that two events A and B are independent iff P(A ∩ B) = P(A) P(______).
The formal definition of independence states that two events A and B are independent iff P(A ∩ B) = P(A) P(______).
In Bayes' formula, we can find P(B2 / A) using the formula P(B2) P(A / ______) / P(A).
In Bayes' formula, we can find P(B2 / A) using the formula P(B2) P(A / ______) / P(A).
The tree diagram describes the process and gives the probability of each ______ of the tree.
The tree diagram describes the process and gives the probability of each ______ of the tree.
Events A and B are dependent if the occurrence of one affects the ______ of the other.
Events A and B are dependent if the occurrence of one affects the ______ of the other.
If A is the event of getting heads on the first two tosses and B is the event of getting tails on the third toss, A and B are ______.
If A is the event of getting heads on the first two tosses and B is the event of getting tails on the third toss, A and B are ______.
In the example of tossing a fair coin three times, for events A and B to be independent, the occurrence of ______ must not influence the occurrence of the other.
In the example of tossing a fair coin three times, for events A and B to be independent, the occurrence of ______ must not influence the occurrence of the other.
Flashcards
Sample Space
Sample Space
The set of all possible outcomes of a random experiment.
Outcome
Outcome
A single trial of a random experiment. It can be a specific outcome like heads (H) or tails (T) when flipping a coin.
Random Experiment
Random Experiment
Represents an experiment that has a number of possible outcomes. It can be something as simple as flipping a coin or a more complex experiment like rolling a die.
Sample Space
Sample Space
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Sample Space Example 1
Sample Space Example 1
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Sample Space Example 2
Sample Space Example 2
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Sample Space Example 3
Sample Space Example 3
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Sample Space Example 4
Sample Space Example 4
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Classical Probability
Classical Probability
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Mutually Exclusive Events
Mutually Exclusive Events
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Classical Probability Formula
Classical Probability Formula
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Relative Frequency Probability
Relative Frequency Probability
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Relative Frequency
Relative Frequency
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Repeated Trials
Repeated Trials
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Trial
Trial
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Conditional Probability
Conditional Probability
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Partition of Sample Space
Partition of Sample Space
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Law of Total Probability
Law of Total Probability
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Joint Probability
Joint Probability
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Bayes' Theorem
Bayes' Theorem
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Chain Rule of Probability
Chain Rule of Probability
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Independent Events
Independent Events
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Dependent Events
Dependent Events
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Probability of an event
Probability of an event
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Probability of an event (formula)
Probability of an event (formula)
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Intersection of events (A ∩ B)
Intersection of events (A ∩ B)
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Union of events (A ∪ B)
Union of events (A ∪ B)
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Probability of union of events
Probability of union of events
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Fair die
Fair die
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Axiomatic approach to probability
Axiomatic approach to probability
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Probability Formula for Independent Events
Probability Formula for Independent Events
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Tree Diagram
Tree Diagram
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Bayes' Formula
Bayes' Formula
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Independence Test
Independence Test
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Probability Formula
Probability Formula
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Product Rule for Independent Events
Product Rule for Independent Events
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Sum Rule for Two Events
Sum Rule for Two Events
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Independence of Events
Independence of Events
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Study Notes
Chapter 2: Probability
- Probability theory originated in gambling in the 18th century and has since expanded into various fields.
- Probability is frequently used in communication, as seen in medical prognoses (e.g., 50-50 chance).
- Probability is a number between 0 and 1, indicating likelihood. Closer to 1 = more likely; closer to 0 = less likely; An event that's impossible = 0; a certain event = 1.
Basic Definitions
- Experiment: A process that yields an observable outcome. A trial is one performance of the experiment.
- Outcome: The specific result of an experiment.
- Sample Space (S): The set of all possible outcomes of a random experiment. One and only one outcome will occur on any given trial of an experiment.
Example 2.1
- An experiment of tossing two coins has possible outcomes recorded as a sample space: {HH, HT, TH, TT}
Example 2.2
- If tossing a coin until a head appears then the sample space could be positive integers: {1, 2, 3...}
Example 2.3
- A sample space for a light bulb's lifespan could be nonnegative real numbers (e.g., time in hours).
- The time being measured only to the nearest hour: {0, 1, 2, 3...}
Sample Space
- Finite: Consists of a finite number of outcomes.
- Discrete: Finite or countably infinite (can be put into one-to-one correspondence with the positive integers)
- Countably Infinite: Has a infinite number of outcomes that can be counted.
- Continuous: Contains an uncountable infinite number of outcomes (represented by a continuum, such as all points on a line).
Event
- A subset of the sample space. Events can consist of more than one outcome
Occurrence of an Event
- An event has occurred if the outcome of the experiment belongs to the set defining the event.
Intersection of Events (A∩B)
- The outcomes that belong to both event A and event B.
Mutually Exclusive Events
- Two events that cannot occur at the same time.
Complementary Event (A’)
- The set of outcomes not in event A .
Equally Likely Events
- Outcomes of a random experiment that have the same chance of occurring.
Classical Probability
- Probability understood in terms of the ratio of favorable outcomes to possible outcomes (if each outcome is equally likely).
- The formula: P(A) = n(A)/n(S), where n(A) is the number of favorable outcomes and n(S) is the total number of possible outcomes.
- Outcomes must be mutually exclusive and equally likely
Examples of Probability Calculations
- Examples are provided demonstrating calculations for events involving dice rolls, coin flips, and other scenarios.
Axiomatic Approach to Probability
- Probability defined using three axioms:
- P(A) >= 0 for all events A.
- P(S) = 1, where S is the sample space.
- If A1, A2, A3,... is a sequence of mutually exclusive events, then P(A1∪A2∪A3...) = P(A1) + P(A2) + P(A3) + ...
Conditional Probability
- Probability of an event given another event has occurred.
- P(B/A) = P(A∩B) / P(A)
Multiplication Rule
- If events A and B are independent, P(A∩B) = P(A)*P(B)
Total Probability
- P(A) = Σ [P(Bi) * P(A/Bi)], where the Bi form a partition of the sample space.
Bayes' Theorem
- Calculating the probability of an event occurring given some conditions.
- P(B/A) = [P(B).P(A/B)] / P(A)
Independent Events
- Two events are independent if the occurrence of one does not affect the probability of the other.
Other Theorems and Rules
- Various theorems and rules are introduced for calculating probabilities of different types of events (e.g., the addition rule for events if they are not mutually exclusive)
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Description
Test your understanding of probability concepts as introduced in Chapter 2. This quiz covers essential definitions, experiments, and sample spaces, providing a solid foundation in probability theory. Enhance your grasp of likelihoods and outcomes with practical examples.
