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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

  1. Experiment: An action or process with uncertain outcomes.
    Example: Tossing a coin.

  2. Sample Space ($S$): The set of all possible outcomes.
    Example: For a coin toss, $S = { \text{Heads, Tails} }$.

  3. Event: A subset of the sample space representing a specific outcome or combination of outcomes.
    Example: Getting heads in a coin toss.

  4. 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

  1. Range: Probability always lies between 0 and 1:

    $$0 \leq P(A) \leq 1$$

  2. Sum of Probabilities:

    The sum of probabilities of all possible outcomes equals 1:

    $$\sum P(A_i) = 1$$

  3. 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

  1. Games of Chance:

    • Calculating odds in card games or dice rolls.

  2. Weather Forecasting:

    • Predicting the likelihood of rain or snow.

  3. Finance:

    • Assessing investment risks and returns.

  4. 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