Statistics and Probability Quiz

Questions and Answers

What is the probability of obtaining a mark of 90 based on the given frequency distribution?

0.123

0.1

0.085

0.079 (correct)

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?

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.

number of outcomes in A Signup and view all the answers

Match the following concepts with their definitions:

Probability = The chance of an event occurring Equally Likely Outcomes = Each outcome has the same chance of occurring Relative Frequency = The ratio of the number of times an event occurs to the total number of trials Mutually Exclusive Events = Events that cannot occur at the same time Signup and view all the answers

What is the probability of not rolling a 6 on a fair die?

$\frac{5}{6}$ Signup and view all the answers

The probability of an impossible event is equal to 1.

False Signup and view all the answers

If events A, B, and C are such that A and C have no overlap, what does this imply about P(A ∩ C)?

P(A ∩ C) = 0 Signup and view all the answers

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) = ___ .

0 Signup and view all the answers

Match the following properties of probability with their descriptions:

P(φ) = Probability of an impossible event P(A ∪ B) = Sum of probabilities minus intersection P(A ∩ B) = Probability of both events occurring P(A' ∩ B') = Probability of neither event occurring Signup and view all the answers

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.

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.