Exploring the probability of combined events through diagrams such as sample space, Venn, and tree diagrams offers a visual and effective way to understand and calculate complex probabilities. These tools simplify the process by breaking down events into more manageable parts, enabling precise probability calculation.
Sample Space Diagrams
Introduction
Sample space diagrams provide a visual representation of all possible outcomes in an event or experiment, crucial for determining the sample space and calculating probabilities.
Example: Rolling Two Dice
Consider rolling two six-sided dice. The sample space diagram for this experiment illustrates all possible outcomes.
- Each die has six faces, numbered 1 to 6.
- The total number of outcomes is .
Representation:
Calculating the probability of rolling a sum of 7:
1. List all outcomes where the sum is 7:
2. Calculate the probability:
Venn Diagrams
Introduction
Venn diagrams show sets and their relationships, ideal for understanding events involving union, intersection, or complement.
Example: Red and Blue Balls
A bag contains 5 red balls and 3 blue balls. The probability of drawing a red or blue ball is represented with a Venn diagram.
- The universal set (U) represents all balls.
- Circles represent the sets of red balls (R) and blue balls (B).
Calculating the probability of drawing a red or a blue ball (R U B):
Given that all balls are either red or blue:
Tree Diagrams
Introduction
Tree diagrams illustrate all possible outcomes of a sequence of events, useful for sequential and dependent events.
Example: Coin Toss and Dice Roll
A coin is tossed, followed by rolling a six-sided die. The tree diagram shows all outcomes.
- First Stage (Coin Toss): Heads (H) or Tails (T).
- Second Stage (Dice Roll): Numbers 1 through 6 from each outcome of the coin toss.
Calculating the probability of getting a Head followed by rolling a 4:
Given :
