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What is the probability of obtaining a mark of 90 based on the given frequency distribution?
What is the probability of obtaining a mark of 90 based on the given frequency distribution?
The probability of obtaining a mark less than or equal to 83 is 0.31.
The probability of obtaining a mark less than or equal to 83 is 0.31.
True
If there are 5 balls in a box with 3 red and 2 black, what is the probability of selecting a black ball?
If there are 5 balls in a box with 3 red and 2 black, what is the probability of selecting a black ball?
0.4
In an equally likely model, the probability of an event A can be calculated using the formula P(A) = _____ / total number of outcomes.
In an equally likely model, the probability of an event A can be calculated using the formula P(A) = _____ / total number of outcomes.
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Match the following concepts with their definitions:
Match the following concepts with their definitions:
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What is the probability of not rolling a 6 on a fair die?
What is the probability of not rolling a 6 on a fair die?
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The probability of an impossible event is equal to 1.
The probability of an impossible event is equal to 1.
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If events A, B, and C are such that A and C have no overlap, what does this imply about P(A ∩ C)?
If events A, B, and C are such that A and C have no overlap, what does this imply about P(A ∩ C)?
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For mutually exclusive events A and B, P(A ∪ B) is equal to the sum of their probabilities: P(A) + P(B). Therefore, P(A ∩ B) = ___ .
For mutually exclusive events A and B, P(A ∪ B) is equal to the sum of their probabilities: P(A) + P(B). Therefore, P(A ∩ B) = ___ .
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Match the following properties of probability with their descriptions:
Match the following properties of probability with their descriptions:
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Study Notes
Frequency Distribution and Relative Frequency
- A frequency distribution table details the grades in a Statistics class, indicating the number of occurrences for each mark.
- Example marks are recorded, ranging from 60 to 99 with their corresponding frequencies.
- The relative frequency of obtaining a mark of 90 is calculated as 5 divided by 63, resulting in approximately 0.079.
- To compute the relative frequency for marks less than or equal to 83, sum the frequencies from 60 to 83, yielding a relative frequency of approximately 0.31.
Classical Definition of Probability
- In an equally likely scenario for a sample space S, the probability of an event A is determined by the formula:
P(A) = Number of outcomes in A / Total number of outcomes. - This formula can also be expressed as:
P(A) = n(A) / n(S), where n(A) represents the number of ways event A can occur, and n(S) represents the total outcomes in the sample space.
Example Scenario
- In a box containing 5 balls (3 red, 2 black), the probability of selecting a red ball is 3 out of 5, or 0.6.
- If rolling a die, the chance of not rolling a 6 can be calculated as 1 - (1/6) = 5/6.
Probability Properties
- The probability of an empty set (impossible event), denoted P(φ), equals 0.
- For events A and B, the union (addition rule) is represented as:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B). - If event A is a subset of B (A ⊆ B), then P(A) is less than or equal to P(B).
Theorems in Probability
- For mutually exclusive events A and B, the intersection probability P(A ∩ B) equals 0, simplifying the union to:
P(A ∪ B) = P(A) + P(B). - For events A, B, and C, the probability of the union is expressed as:
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C).
Worked Example of Probabilities
- Given P(A) = 1/2, P(B) = 3/1, P(C) = 1/4, with A ∩ C = φ, B ∩ C = φ, and P(A ∩ B) = 6/1:
- Calculate P[(A ∩ B)'] as 1 - P(A ∩ B) = 5/6.
- For P(A ∩ B'), use the relation where A = (A ∩ B') ∪ (A ∩ B) to find P(A ∩ B').
- Determine P[(A ∪ B)'] and P(A' ∩ B') using similar calculations.
Axioms of Probability
- Axiom 1 states probabilities must fall between 0 and 1 (0 ≤ P(A) ≤ 1).
- Axiom 2 asserts P(S) = 1 and P(φ) = 0; the outcome will be among the simple outcomes.
- Axiom 3 states that for mutually exclusive events A and B:
P(A ∪ B) = P(A) + P(B).
Infinite Sample Spaces
- For an infinite sequence of mutually exclusive events:
P(A1 ∪ A2 ∪ A3 ∪ ...) = ΣP(Ai) from i=1 to ∞.
Elementary Properties of Probability
- Property 1 defines the probability of the complement of an event:
P(Ā) = 1 - P(A), indicating the probability that event A does not occur.
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Description
Test your understanding of frequency distribution and the classical definition of probability. This quiz covers concepts like relative frequency calculation and the basic probability formula, helping you grasp essential statistical principles.
