Conditional Probability – GCSE Maths

Introduction

Conditional Probability is the probability of an event occurring given that another event has already occurred.
Studying of Conditional Probability is important because it helps us understand how the probability of an event changes when we know that another event has occurred.
This concept is essential in real-world situations where outcomes are not independent.

What is Conditional Probability?

We know, if one event depends upon the outcome of another event, the two events are Dependent events.
A Conditional Probability is the probability of a dependent event in which probability of the second outcome depends on what has already happened in the first outcome.

Example:

If there is a bag with red and blue balls. Picking one ball out and don’t put it back, then take another one, the chance of getting a red or blue ball on the second draw depends on what happened first.

How to Calculate Conditional Probability using Tree Diagrams?

A Tree Diagram can be used to solve Conditional Probability using dependent events

Steps to solve conditional Probability using Tree Diagram:

Step #1: Draw the Branch and label the probabilities.
Step #2: Add Dependent Branches
Step #3: Apply the Condition
Step #4: Find the Probability

Solved Example:

Problem: Ivan has a Bag with 3 red and 2 green marbles. He picks 2 marbles without replacement. What’s the probability the second marble is red given the first was green?

Solution:

Step #1: Draw the Branch and label the probabilities.

Possible outcomes for first pick:

- Total Marbles = 5
- Red (3 out of 5 marbles) = 3/5
- Green (2 out of 5 marbles) = 2/5

Step #2: Add Dependent Branches

If first was Red:

- Remaining marbles: 2 red, 2 green
- Next pick will be:

If first was Green:

- Remaining marbles: 3 red, 1 green
- Next pick will be:

Step #3: Apply the Condition

First marble was green, so only follow the green path.

Step #4: Find the Probability

On the Green path, the chance the second marble is red is 3/4

Final Answer: The probability the second marble is red is 3/4

Use Two-way Table to Calculate Conditional Probability

A Two-way table is a table shows how often different combinations of two events happen together.

Steps to solve conditional Probability using Two-way table:

Step #1: Create the Two-Way Table
Step #2: Apply the Condition
Step #3: Find the Probability

Solved Example:

Problem: If a random cheesy pizza slice is picked from

6 pepperonis (with cheese)
3 olive pizzas (with cheese)

What’s the probability it’s pepperoni with cheese?

Solution:

Step #1: Create the Two-Way Table

Total Pizza Slices = 9

Step #2: Apply the Condition

- The Condition is pepperoni with cheese slice.

Step #3: Find the Probability

Using the Table:

The probability of the cheesy slice is pepperoni is 6/9

Final Answer: The probability of the cheesy slice is pepperoni is 6/9

Solved Example:

Problem: A bag contains:

You randomly pick one ball, don’t put it back, then pick a second ball. What’s the probability the second ball is red, given that the first ball was blue?

Solution:

Step #1: Create the Two-Way Table

If the first ball was blue then,

- Total number of balls left: 10 – 1 = 9

Step #2: Apply the Condition

- The condition is that if the first ball picked is blue, then the second ball is red.

Step #3: Find the Probability

Using the Table:

The probability of the second ball is red if the first was blue 4/9

Final Answer: The probability of the second ball is red if the first was blue 4/9

Solved Example:

Problem: A standard deck has 52 cards. You draw 2 cards without replacement. What’s the probability the second card is red, given the first card was black?

Solution:

Step #1: Draw the Branch and label the probabilities.