Probability Calculator

What is Probability?

Probability is a measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Formulas and Their Meanings

1. Single Event Probability: \[P(A) = \frac{n(A)}{n}\] Where \(n(A)\) is the number of favorable outcomes and \(n\) is the total number of possible outcomes.

2. Probability of Event Not Occurring: \[P(A') = 1 - P(A)\] This is the complement of the probability of A occurring.

3. Probability of Two Events Occurring (Intersection): \[P(A \cap B) = \frac{n(A) + n(B) - n}{n}\] This represents the probability of both events A and B occurring.

4. Probability of Either Event Occurring (Union): \[P(A \cup B) = \frac{n(A) + n(B)}{n}\] This represents the probability of either event A or B (or both) occurring.

5. Conditional Probability: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\] This is the probability of event A occurring, given that event B has occurred.

Calculation Steps

1. Identify the total number of possible outcomes (n).
2. Determine the number of favorable outcomes for each event (n(A), n(B)).
3. Calculate the individual probabilities using the formulas above.
4. For combined probabilities, use the intersection and union formulas.
5. For conditional probability, use the given formula if applicable.

Example Calculation

Let's calculate probabilities for a standard deck of 52 cards:

1. P(drawing a heart) = 13/52 = 1/4 = 0.25
2. P(not drawing a heart) = 1 - 0.25 = 0.75
3. P(drawing a king) = 4/52 = 1/13 ≈ 0.0769
4. P(drawing a heart or a king) = (13 + 4) / 52 = 17/52 ≈ 0.3269
5. P(drawing a king given it's a heart) = (1/52) / (13/52) = 1/13 ≈ 0.0769

Visual Representation

This pie chart represents the probability of drawing a heart (0.25) versus other suits (0.75) from a standard deck of cards.