Probability
Random Experiments and Outcomes
What is Probability?
Probability is the branch of mathematics that measures how likely an event is to occur. It quantifies uncertainty and helps us make predictions about random events.
Where Does the Word Come From?
The word "probability" comes from the Latin word "probabilis" meaning "provable" or "worthy of approval."
A Brief History of Probability:
| Time Period | Development | Key Contributors |
|---|---|---|
| 16th-17th Century | Formal study began | Girolamo Cardano (Italy) |
| 1654 | Birth of probability theory | Blaise Pascal and Pierre de Fermat (France) |
| 18th Century | Law of Large Numbers | Jacob Bernoulli (Switzerland) |
| 20th Century | Modern probability theory | Andrey Kolmogorov (Russia) |
The Gambler's Problem That Started It All:
In 1654, a gambler named Chevalier de Méré asked Pascal: "Why do I lose money betting on getting at least one six in 4 rolls of a die, but win betting on at least one double-six in 24 rolls of two dice?" This question led Pascal and Fermat to develop probability theory.
Scope and Applications of Probability:
| Field | Application Example |
|---|---|
| **Medicine** | Probability of a treatment being effective |
| **Insurance** | Calculating risk and setting premiums |
| **Gaming** | Designing fair games and predicting odds |
| **Sports** | Probability of a team winning |
| **Finance** | Stock market predictions |
| **Artificial Intelligence** | Machine learning algorithms |
Meaning of Probability:
- Probability is always a number between 0 and 1
- 0 means the event is impossible (will never happen)
- 1 means the event is certain (will definitely happen)
- The closer the probability is to 1, the more likely the event
- Outcome = one result; sample space = all outcomes.
- An event is a set of outcomes of interest.
Empirical Probability
What is Experimental Probability?
Experimental probability (also called empirical probability) is the probability determined by actually performing an experiment and recording the results. It is based on observed data, not theoretical calculations.
Theoretical vs. Experimental Probability:
| Aspect | Theoretical Probability | Experimental Probability |
|---|---|---|
| **Calculation** | Favorable outcomes / Total possible outcomes | Observed frequency / Total trials |
| **Accuracy** | Exact, fixed value | Approximate, changes with trials |
| **Example** | P(heads) = 1/2 | After 100 flips, 48 heads → 0.48 |
Formula for Experimental Probability:
\[
P(\text{Event}) = \frac{\text{Number of times event occurred}}{\text{Total number of trials}}
\]
Law of Large Numbers:
As the number of trials increases, the experimental probability gets closer to the theoretical probability.
Why Use Experimental Probability?
- When theoretical probability is unknown or difficult to calculate
- To test if a game or process is fair
- In real-world situations where outcomes are not equally likely
Steps to Find Experimental Probability:
- Perform the experiment many times (trials)
- Count how many times the event occurs
- Divide the count by the total number of trials
- P(E) = favourable ÷ total, between 0 and 1.
- P(not E) = 1 − P(E); impossible = 0, certain = 1.
Coins, Dice and Cards
What is a Single Event?
A single event is one outcome or a set of outcomes from a single experiment. For example, rolling a die once and getting an even number is a single event.
Types of Events:
| Event Type | Definition | Example |
|---|---|---|
| **Compound Event** | Two or more outcomes combined | Rolling an odd number (1,3,5) |
| **Impossible Event** | Cannot occur | Rolling a 7 |
| **Certain Event** | Will always occur | Rolling a number < 7 |
Basic Probability Formula:
\[
P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
\]
Key Properties:
- 0 ≤ P(E) ≤ 1 for any event E
- P(impossible event) = 0
- P(certain event) = 1
- Sum of probabilities of all elementary events = 1
Common Probability Problems:
| Experiment | Sample Space | Example Event |
|---|---|---|
| Rolling a die | {1,2,3,4,5,6} | Getting a number > 4 |
| Drawing a card | 52 cards | Drawing a heart |
| Spinning a spinner | Equal sections | Landing on red |
- Die: 6 outcomes; two dice: 36; deck: 52 cards.
- P(E) = favourable ÷ total for equally likely outcomes.