Basic Probability Concepts

Concept Explanation

Probability measures the likelihood that an event will occur. It is expressed as a number between 0 and 1, where:

• 0 means the event is impossible
• 1 means the event is certain
• Values between 0 and 1 indicate varying degrees of likelihood

The basic formula for probability is: Probability = Number of favorable outcomes ÷ Total number of possible outcomes

Probability can be expressed as a fraction, decimal, or percentage. For example, a probability of 1/4 can also be written as 0.25 or 25%.

Worked Examples

Example 1

Problem: A bag contains 3 red marbles, 2 blue marbles, and 5 green marbles. What is the probability of randomly selecting a blue marble?
Solution: 2/10 = 1/5 = 0.2 = 20%
Explanation: There are 2 favorable outcomes (blue marbles) out of 10 total possible outcomes (all marbles). So, P(blue) = 2/10 = 1/5.

Example 2

Problem: A spinner has 4 equal sections colored red, blue, yellow, and green. What is the probability of the spinner landing on yellow?
Solution: 1/4 = 0.25 = 25%
Explanation: There is 1 favorable outcome (yellow section) out of 4 total possible outcomes (all sections). So, P(yellow) = 1/4.

Example 3

Problem: A standard six-sided die is rolled. What is the probability of rolling an even number?
Solution: 3/6 = 1/2 = 0.5 = 50%
Explanation: There are 3 favorable outcomes (2, 4, 6) out of 6 total possible outcomes (1, 2, 3, 4, 5, 6). So, P(even) = 3/6 = 1/2.

Common Errors

Practice Problems

1. Problem:

   A bag contains 4 red balls and 6 blue balls. What is the probability of randomly selecting a red ball?

   Solution: 4/10 = 2/5 = 0.4 = 40%
2. Problem: A standard deck has 52 cards. What is the probability of drawing a heart?
   Solution: 13/52 = 1/4 = 0.25 = 25%
3. Problem: If you roll a six-sided die, what is the probability of rolling a number greater than 4?
   Solution: 2/6 = 1/3 = 0.33 = 33.3%
4. Problem:

   A spinner has 5 equal sections numbered 1 through 5. What is the probability of spinning an odd number?

   Solution: 3/5 = 0.6 = 60%
5. Problem: If you flip a fair coin twice, what is the probability of getting heads both times?
   Solution: 1/4 = 0.25 = 25%

Real-World Application Example

Probability is used in many real-world contexts, from weather forecasting (“30% chance of rain”) to games of chance, insurance calculations, and medical diagnoses. Understanding probability helps us make informed decisions in uncertain situations, assess risks, and interpret statistical information in news and research. For example, knowing the probability of different outcomes helps doctors recommend treatments, investors manage portfolios, and sports analysts predict game results.