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Event Relations Calculator

Calculate probabilities, unions, intersections, and conditional probabilities

Event A Favorable Outcomes

Event B Favorable Outcomes

Both Events (A ∩ B) Outcomes

Total Possible Outcomes

Core Concepts

Random Experiments & Sample Spaces

Random Experiment

An experiment that satisfies three conditions: repeatability (can be repeated under identical conditions), uncertainty (outcome cannot be predicted beforehand), and exhaustive (all possible outcomes can be listed).

Examples: Tossing a coin (Ω = {H, T}), rolling a dice (Ω = {1, 2, 3, 4, 5, 6}), drawing a card from deck (52 outcomes)

Sample Space (Ω)

The set of all possible outcomes of a random experiment. Contains every possible outcome, no outcome appears twice, outcomes are mutually exclusive, and can be finite, countably infinite, or uncountably infinite.

Events & Operations

Event Operations

Union (A ∪ B)

A \cup B

Event that occurs when A or B (or both) occur

Intersection (A ∩ B)

A \cap B

Event that occurs when both A and B occur

Complement (Ā)

\overline{A}

Event that occurs when A does not occur

Probability Definitions

Classical Probability

P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of equally likely outcomes}}

Requires finite sample space, all outcomes equally likely, and known structure of experiment.

Statistical Probability

P(A) = \lim_{n \to \infty} \frac{n_A}{n}

Based on empirical observations, requires large number of trials, frequency stabilizes as n increases.

Geometric Probability

P(A) = \frac{\text{Measure of favorable region}}{\text{Measure of total region}}

For continuous sample spaces with uniform distribution, uses geometric interpretation with area ratios.

Probability Laws & Properties

Axioms of Probability

1. Non-negativity

P(A) \geq 0 \text{ for all events } A

Probability is never negative

2. Normalization

P(\Omega) = 1

Probability of certain event is 1

3. Countable Additivity

P(\bigcup_{i=1}^\infty A_i) = \sum_{i=1}^\infty P(A_i) \text{ if } A_i \text{ are mutually exclusive}

Probability of union equals sum for exclusive events

Fundamental Properties

Complement Rule

P(\overline{A}) = 1 - P(A)

Addition Rule

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Monotonicity

\text{If } A \subseteq B, \text{ then } P(A) \leq P(B)

Worked Examples

Example 1: Classical Probability

Problem:

From a standard 52-card deck, find the probability of drawing a red king or a black ace.

Solution:

1. Identify events: A = {red king}, B = {black ace}
2. Check if mutually exclusive: Yes, no overlap
3. Count favorable outcomes: |A| = 2, |B| = 2
4. Apply addition rule: P(A ∪ B) = P(A) + P(B) = 2/52 + 2/52 = 4/52 = 1/13

Example 2: Complement Rule

Problem:

A system has 5 independent components, each working with probability 0.9. Find probability system works if at least 3 must work.

Solution:

1. Use complement: P(at least 3) = 1 - P(at most 2)
2. Calculate P(exactly 0) = C(5,0)(0.1)⁵ = 0.00001
3. Calculate P(exactly 1) = C(5,1)(0.1)⁴(0.9) = 0.00045
4. Calculate P(exactly 2) = C(5,2)(0.1)³(0.9)² = 0.0081
5. Sum: P(at most 2) = 0.00856
6. Answer: P(at least 3) = 1 - 0.00856 = 0.99144

Practice Quiz

10

Questions

0

Correct

0%

Accuracy

1

What is the sample space for rolling a fair six-sided die?

Not attempted

2

If

P(A) = 0.3

and

P(B) = 0.5

, and

A

and

B

are mutually exclusive, what is

P(A \cup B)

?

Not attempted

3

What is the complement rule formula?

Not attempted

4

If

P(A \cap B) = 0.2

,

P(A) = 0.4

, and

P(B) = 0.5

, are events

A

and

B

independent?

Not attempted

5

What is the formula for conditional probability

P(A|B)

?

Not attempted

6

In classical probability, what condition must be satisfied?

Not attempted

7

If

P(A) = 0.6

, what is

P(\overline{A})

?

Not attempted

8

What does the addition rule state for two events

A

and

B

?

Not attempted

9

Which of the following is an axiom of probability?

Not attempted

10

If events

A

and

B

are independent, which of the following is true?

Not attempted

Frequently Asked Questions

What is the difference between classical and statistical probability?

Classical probability assumes all outcomes are equally likely and uses the ratio of favorable to total outcomes. Statistical probability is based on observed frequencies from repeated experiments and approaches the true probability as the number of trials increases.

When should I use the complement rule?

Use the complement rule when calculating the probability of 'at least' or 'at most' events is easier by finding the probability of the complement. For example, P(at least one) = 1 - P(none).

How do I know if two events are independent?

Two events A and B are independent if P(A ∩ B) = P(A) × P(B). This means the occurrence of one event does not affect the probability of the other. Mutually exclusive events are NOT independent (unless one has probability 0).

What is the difference between mutually exclusive and independent events?

Mutually exclusive events cannot occur simultaneously (P(A ∩ B) = 0), while independent events can occur together and the occurrence of one doesn't affect the other (P(A ∩ B) = P(A) × P(B)). These are different concepts and should not be confused.

How do I apply probability to real-world problems?

Start by identifying the sample space and defining events clearly. Choose the appropriate probability definition (classical, statistical, or geometric) based on the problem context. Use probability laws and properties systematically, and always verify your answer makes intuitive sense.

Probability Theory Fundamentals

Core Concepts

Random Experiment

Sample Space (Ω)

Event Operations

Probability Definitions

Probability Laws & Properties

1. Non-negativity

2. Normalization

3. Countable Additivity

Complement Rule

Addition Rule

Monotonicity

Worked Examples

Frequently Asked Questions