Understand the fundamental concepts of probability and how to calculate it.
Example 1: Basic Probability Concepts
Learn about experiments, outcomes, sample space, and events.
Experiment: An action or process with an uncertain result (e.g., tossing a coin, rolling a die).
Outcome: A single possible result of an experiment (e.g., getting 'Heads', rolling a '4').
Sample Space (S): The set of all possible outcomes of an experiment (e.g., for a coin toss, S = {Heads, Tails}).
Event (E): A specific outcome or a set of outcomes from the sample space that we are interested in (e.g., getting an even number when rolling a die, E = {2, 4, 6}).
Example 2: Calculating Probability of a Single Event
How to calculate the probability of an event occurring. Example: Probability of rolling a '3' on a fair six-sided die.
Formula: The probability of an event E, denoted P(E), is calculated as:
P(E) = (Number of favorable outcomes for E) / (Total number of possible outcomes in S)
Step 1: Identify the total number of possible outcomes (size of sample space). For a fair die, S = {1, 2, 3, 4, 5, 6}, so Total Outcomes = 6.
Step 2: Identify the number of favorable outcomes for the event. Event E = rolling a '3'. There is only one '3' on the die. So, Favorable Outcomes = 1.
Step 3: Calculate the probability. P(rolling a '3') = 1 / 6.
Note: Probability values always range from 0 (impossible event) to 1 (certain event).
Example 3: Probability of 'Not' an Event (Complement)
Understand complementary events. Example: If P(rain) = 0.3, what is P(not rain)?
Complementary Event: The complement of an event E, denoted E' or Ec (or "not E"), consists of all outcomes in the sample space that are not in E.
Formula: The sum of the probability of an event and its complement is always 1. So, P(E) + P(not E) = 1.
Example 4: Independent vs. Dependent Events (Introduction)
Briefly explain the difference between independent and dependent events.
Independent Events: Two events are independent if the occurrence of one event does not affect the probability of the other event occurring.
Example: Tossing a coin twice. The outcome of the first toss does not affect the outcome of the second toss. P(A and B) = P(A) * P(B) for independent events.
Dependent Events: Two events are dependent if the occurrence of one event changes the probability of the other event occurring.
Example: Drawing two cards from a deck without replacement. The outcome of the first draw affects the probabilities for the second draw. P(A and B) = P(A) * P(B|A) for dependent events (where P(B|A) is the conditional probability of B given A).
Practice Mode
Enter a simple probability question to see its calculation.
Note: This basic solver can handle simple questions like "Probability of getting a 5 on a die roll" or "P(Heads) on a coin toss". It understands keywords like 'die', 'coin', 'heads', 'tails', specific numbers on a die, or simple fractions like '1/6'.
Related Concepts
Explore these related mathematical concepts to deepen your understanding of probability.
Experiment
A process that leads to well-defined outcomes.
Outcome
A possible result of an experiment.
Sample Space
The set of all possible outcomes of an experiment.
Event
A subset of the sample space; one or more outcomes.
Theoretical Probability
Probability based on reasoning or calculation (favorable/total).
Empirical Probability
Probability based on observed frequencies from experiments.
Complementary Events
Two events where one or the other must happen, but they cannot happen together.
Mutually Exclusive Events
Events that cannot occur at the same time.
Independent Events
Occurrence of one event does not affect the probability of another.
Dependent Events
Occurrence of one event affects the probability of another.
Conditional Probability
Probability of an event occurring, given that another event has already occurred.
Random Variable
A variable whose value is a numerical outcome of a random phenomenon.