Probability Basics

Probability is the branch of mathematics that measures the likelihood of an event occurring. It provides a foundation for statistical analysis and decision-making under uncertainty.

What is Probability?

Probability quantifies uncertainty, representing it as a number between 0 and 1:

0: The event will not happen (impossible).
1: The event will certainly happen.

Formula:

The probability of an event $A$ is given by:

\[P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\]

Key Terminology

Experiment: An action or process with uncertain outcomes.
Example: Tossing a coin.
Sample Space ($S$): The set of all possible outcomes.
Example: For a coin toss, $S = { \text{Heads, Tails} }$.
Event: A subset of the sample space representing a specific outcome or combination of outcomes.
Example: Getting heads in a coin toss.
Favorable Outcome: The specific outcome(s) of interest in an event.
Example: Rolling a 6 on a die.

Types of Probability

1. Theoretical Probability

Based on logical analysis without actual experiments.

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

Example: The probability of rolling a 4 on a fair die is:

\[P(4) = \frac{1}{6}\]

2. Experimental Probability

Based on the results of experiments or observations.

\[P(A) = \frac{\text{Number of times event occurs}}{\text{Total number of trials}}\]

Example: If a coin lands on heads 7 times in 10 tosses, the probability of heads is:

\[P(\text{Heads}) = \frac{7}{10}\]

3. Subjective Probability

Based on intuition, personal judgment, or experience.

Example: Estimating the probability of rain tomorrow based on current weather patterns.

Properties of Probability

Range: Probability always lies between 0 and 1:
\[0 \leq P(A) \leq 1\]
Sum of Probabilities:

The sum of probabilities of all possible outcomes equals 1:
\[\sum P(A_i) = 1\]
Complement Rule:
The probability of an event not happening is:
\[P(\text{Not A}) = 1 - P(A)\]
Example: If $P(A) = 0.7$, then $P(\text{Not A}) = 0.3$.

Common Applications of Probability

Games of Chance:
- Calculating odds in card games or dice rolls.
Weather Forecasting:
- Predicting the likelihood of rain or snow.
Finance:
- Assessing investment risks and returns.
Healthcare:
- Estimating the effectiveness of a new drug in clinical trials.

Example Problems

Problem 1: Tossing a Coin

What is the probability of getting heads in a single coin toss?
Solution:

\[P(\text{Heads}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{2}\]

Problem 2: Rolling a Die

What is the probability of rolling an even number on a fair six-sided die?
Solution: Favorable outcomes = ${2, 4, 6}$

\[P(\text{Even}) = \frac{3}{6} = \frac{1}{2}\]

Conclusion

Probability forms the foundation for understanding randomness and uncertainty. By learning its basics, you can analyze real-world situations, make predictions, and make informed decisions.

Next Steps: Rules of Probability