Introduction
- Probability is a branch of mathematics that deals with calculating the likelihood of events occurring.
- Probability helps us measure and predict the chances of events happening in the real world like-
Real-life applications of probability
Probability in medical diagnosis and airplane safety
What is a Probability?
- Probability is a measure of how likely an event is to occur.
Probability Rules:
- Range of Probability – $0 \leq P(E) \leq 1$
- Sum of Probabilities – For all outcomes in $S$, $\sum P(E) = 1$.
- Complement Rule – $P(not E) = 1 – P(E)$.
For an event $E$, probability $P(E)$ is calculated as:
What is an Experimental Probability?
- Experimental probability is the probability of an event based on actual experiments or observations.
- It is calculated by dividing the number of times the event occurs by the total number of trials performed.
- Mathematically,
Need help with Probability?
Our tutors explain it step by step, matched to your exam board.
Book a Consultation →Key Points
- Experiment: An experiment is any process that produces a well-defined outcome.
Examples:
Probability experiments: Dice roll, coin toss, and card draw
- Outcome: An outcome is a possible result of an experiment.
Example: In a coin toss, Possible outcomes are Heads (H) or Tails (T).
Coin flip head and tail sides
- Sample Space (S): The sample space is the set of all possible outcomes of an experiment.
Example: For a die roll,
Sample space of a 6-sided die
Events:
An event is any subset of the sample space. It can include,
- Simple Event: A single outcome (e.g., rolling a 3).
- Compound Event: Multiple outcomes (e.g., rolling an even number $\{2, 4, 6\}$).
Solved Examples
Total number of marbles:
Probability of picking a blue marble:
Final Answer: $\frac{3}{10}$
- Successful outcomes (Heads): $6$
- Total flips: $10$
The experimental probability of getting Heads is $\frac{6}{10}$.
Final Answer: $\frac{6}{10}$
The event is getting heads, where:
- Heads occurred $3$ times.
- Total number of coin flips = $4$.
Using the Experimental Formula:
So, the Experimental probability of getting heads is $0.75$.
Final Answer: $0.75$
Given:
- Total Balls = $5 + 3 = 8$
- Blue balls = $3$
Use the complement rule:
The probability that the ball is not blue is $\frac{5}{8}$.
Final Answer: $\frac{5}{8}$
Given:
- Total possible outcomes = $2$ (Head, Tail)
Probability of getting a head:
The probability of getting a head $\frac{1}{2}$.
Final Answer: $\frac{1}{2}$
