Lesson 85: Introduction to Probability
Probability is the branch of mathematics that studies how likely an event is to occur. It helps us measure uncertainty and make informed predictions about real-world situations. Probability values always fall between 0 (impossible) and 1 (certain).
Core Ideas
- Sample Space: The set of all possible outcomes.
- Event: A subset of outcomes we are interested in.
- Probability Formula: \( P(\text{event}) = \dfrac{\text{number of favorable outcomes}}{\text{total number of outcomes}} \)
- Theoretical Probability: Based on reasoning and counting outcomes.
- Experimental Probability: Based on actual trials or experiments.
Examples
Example 1 — Coin Toss:
Sample space: {Heads, Tails}. Probability of Heads = \( \dfrac{1}{2} \).
Example 2 — Rolling a Die:
Sample space: {1, 2, 3, 4, 5, 6}. Probability of rolling an even number = \( \dfrac{3}{6} = \dfrac{1}{2} \).
Example 3 — Drawing a Marble:
A bag contains 3 red, 2 blue, and 5 green marbles. Probability of drawing a blue marble = \( \dfrac{2}{10} = \dfrac{1}{5} \).
Practice Problems
- What is the probability of rolling a number greater than 4 on a standard die?
- If you flip a coin 20 times and get 12 heads, what is the experimental probability of heads?
- A bag has 4 black, 3 white, and 3 gray stones. What is the probability of picking a gray stone?