Probability and Events

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Questions and Answers

What is the probability of events A and B being independent?

P(AB) = P(A) / P(B)

P(AB) = P(A) - P(B)

P(AB) = P(A) + P(B)

P(AB) = P(A) * P(B) (correct)

Using Baye's Theorem, which formula represents the conditional probability of event Ai given event X?

P(Ai | X) = P(Ai) P(X)

P(Ai | X) = P(Ai) / (sum of P(Aj))

P(Ai | X) = P(Ai) P(X) / (P(A1) P(X) + P(A2) P(X) + ... + P(An) P(X)) (correct)

P(Ai | X) = P(X) / P(Ai)

In the problem with three boxes, how many total white balls are present across all boxes?

10 (correct)

What is the probability of drawing a red ball from the first box?

3/5 Signup and view all the answers

If events A and B are not independent, which of the following must be true?

P(AB) ≠ P(A) * P(B) Signup and view all the answers

Which box has the highest proportion of red balls?

Second box Signup and view all the answers

What is the probability of drawing a red ball from the third box?

1/3 Signup and view all the answers

In the scenario of mutually exclusive events A1, A2, ..., An, what must the sum of their probabilities equal?

Equal to 1 Signup and view all the answers

What is the probability of an impossible event?

0 Signup and view all the answers

Which statement correctly describes mutually exclusive events?

Their intersection is the empty set. Signup and view all the answers

If the sample space S has 10 outcomes and event A contains 3 of those outcomes, what is the probability P(A)?

0.3 Signup and view all the answers

Which axiom of probability states that the probability of the entire sample space equals 1?

Axiom (ii) Signup and view all the answers

What does the notation P(Ac) represent?

The complement of event A's probability Signup and view all the answers

If P(A) = 0.6, what is the value of P(Ac)?

0.4 Signup and view all the answers

Which of the following properties of probability states that the probability cannot exceed 1?

P(A) ≤ 1 Signup and view all the answers

If events A and B are independent, which property holds true?

P(A ∩ B) = P(A) * P(B) Signup and view all the answers

What is the correct formula for the probability of the union of two events A and B when they are mutually exclusive?

P(A+B) = P(A) + P(B) Signup and view all the answers

If events A, B, and C are mutually exclusive, what is the correct expression for the probability of their union?

P(A+B+C) = P(A) + P(B) + P(C) Signup and view all the answers

What denotes the conditional probability of event B given event A?

P(B|A) Signup and view all the answers

Under what condition are events A and B considered independent?

P(AB) = P(A)P(B) Signup and view all the answers

What is the probability that at least one of two randomly selected watches is defective from a pack of 10 where 3 are defective?

11/15 Signup and view all the answers

In a leap year, how many days does it consist of and how can it provide 53 Sundays?

366 days with a possibility of 1 or 2 Sundays in the extra days Signup and view all the answers

Using the theorem of compound probability, what is the equation for events A and B?

P(AB) = P(A) * P(B|A) Signup and view all the answers

If events A and B are not mutually exclusive, which expression correctly calculates P(A+B)?

P(A+B) = P(A) + P(B) - P(AB) Signup and view all the answers

What is the probability of choosing the second box, A2?

$\frac{1}{3}$ Signup and view all the answers

What is the mathematical expectation E(x) if p1 = $\frac{1}{3}$, x1 = 2, p2 = $\frac{1}{3}$, x2 = 4, and p3 = $\frac{1}{3}$, x3 = 6?

4 Signup and view all the answers

What is the total probability P(X) of drawing a red ball from any box if the probabilities associated with each box are given?

$\frac{62}{135}$ Signup and view all the answers

If the probability of drawing an ace from one pack is $\frac{1}{13}$, what is the probability of drawing at least one ace from two packs?

$\frac{25}{169}$ Signup and view all the answers

What does the variance of a variable x denote mathematically?

E(x-m)2 Signup and view all the answers

What is the relationship between events A and B, when each is independently drawing a card from two distinct packs of cards?

Independent events Signup and view all the answers

How is the probability of event X calculated using Bayes' theorem when given P(A2) and other probabilities?

P(X|A2) * P(A2) / P(X) Signup and view all the answers

What does E(x) represent if it equals m in the context of mathematical expectation?

Mean Signup and view all the answers

Study Notes

Sample Space and Events

Sample Space (S): The set of all possible outcomes of a random experiment.
Event (A): A subset of the sample space.
Event Space: The set of all possible events associated with a random experiment.

Random Experiment

A process with a defined set of possible outcomes where the specific outcome is unpredictable.

Mutually Exclusive Events

Two events that cannot occur simultaneously.
Symbolically: A ∩ B = Φ (empty set) or P(A ∩ B) = 0

Classical Definition of Probability

P(A): The probability of event A occurring.
Formula: P(A) = m(A) / n(S), where:
- m(A) is the number of outcomes favorable to A.
- n(S) is the total number of possible outcomes in the sample space.

Axioms of Probability

Axiom (i): P(A) ≥ 0 (Probability is non-negative)
Axiom (ii): P(S) = 1 (Probability of the certain event is 1)
Axiom (iii): For mutually exclusive events A1, A2, ... An, P(A1 ∪ A2 ∪ ... ∪ An) = P(A1) + P(A2) + ... + P(An)

Probability Notation

P(A): Probability of event A.
P(A̅) or P(Aᶜ): Probability of event A not occurring.
P(A ∪ B) or P(A + B): Probability of at least one of the events A and B occurring.
P(A ∩ B) or P(AB): Probability of both events A and B occurring.
P(A | B): Conditional probability of event A given that event B has already occurred.

Properties of Probability

1. P(φ) = 0: The probability of the impossible event is 0.
2. P(A) ≤ 1: The probability of any event is less than or equal to 1.
3. P(A̅) = 1 − P(A): Probability of an event not occurring is 1 minus the probability of it occurring.
4. P(B ∩ A̅) = P(B) − P(B ∩ A): Useful to calculate probability of B occurring but A not occurring

Theorem of Addition

For any events A and B: P(A + B) = P(A) + P(B) – P(AB).
For mutually exclusive events A and B: P(AB) = 0, so P(A + B) = P(A) + P(B).
For any three events A, B, and C:
- P(A + B + C) = P(A) + P(B) + P(C) – P(BC) – P(CA) – P(AB) + P(ABC).
- If mutually exclusive, P(A + B + C) = P(A) + P(B) + P(C).

Conditional Probability

P(B | A): Probability of event B occurring given that event A has already occurred.
Formula: P(B | A) = P(AB) / P(A), where P(A) > 0.

Theorem of Compound Probability

P(AB): Probability of events A and B occurring simultaneously.
Formula: P(AB) = P(A) * P(B | A).

Independent Events

Two events are independent if the occurrence of one event does not affect the probability of the other event occurring.
Condition: P(AB) = P(A) * P(B).

Baye's Theorem

Used to calculate the conditional probability of an event Ai given that another event X has occurred.
Formula:
- P(Ai | X) = [P(Ai) * P(X | Ai)] / [∑ P(Aj) * P(X | Aj)] (where j goes from 1 to n).
- P(Ai) is the prior probability of Ai.
- P(X | Ai) is the conditional probability of X given Ai.

Mathematical Expectation (E(x))

The expected value of a random variable 'x' with n possible outcomes.
Formula: E(x) = ∑ (pi * xi) (where 'i' denotes the index of the outcome, 'pi' is the probability of outcome 'i', and 'xi' is the value of 'x' for outcome 'i').

Variance (Var(x))

Measures the spread of a random variable around its mean.
Formula: Var(x) = E(x²) - [E(x)]²

Examples

Example 1: Calculating the probability of a leap year having 53 Sundays.
Example 2: Determining if two events are independent given their probabilities.
Example 3: Applying Bayes' Theorem to calculate the probability of a certain event based on given conditions.
Example 4: Determining the probability of at least one specific event occurring from two independent trials.

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Description

This quiz covers fundamental concepts in probability, including sample space, events, and mutually exclusive events. Understand classical definitions and axioms of probability through a variety of questions designed to test your knowledge and application skills.