 # 10.05 Solving problems with probability

Lesson

## Probability calculations

Many situations in probability can be organised into Venn diagrams, arrays, and tree diagrams to organise the information, determine the size of different groups and do calculations.

The formulas and notation encountered can also help us to organise the information given in a question and calculate different probabilities.

### Examples

#### Example 1

Given that P(A \cap B) =0.2 and P(A \cap B')=0.3.

a

What is the value of P(A)?

Worked Solution
Create a strategy

We can use the formula: P(A)= P(A \cap B) + P(A \cap B').

Apply the idea
b

What is the value of P(B), given that the events are independent?

Worked Solution
Create a strategy

We can use the formula: P(B)=\dfrac{P(A \cap B)}{P(A)}.

Apply the idea
c

Given that A and B are independent find P(A \cup B').

Worked Solution
Create a strategy

We can use the formula: P(X \cup Y)=P(X)+ P(Y) - P(X \cap Y).

Apply the idea

#### Example 2

Consider the following probability Venn Diagram:

Find P(A|B).

Worked Solution
Create a strategy

We can use the conditional probability formula: P(A|B) = \dfrac{P(A \cap B)}{P(B)}.

Apply the idea
Idea summary

Probability can be organised into Venn diagrams, arrays, and tree diagrams to organise the information to help solve problems.

We can also use the formulas for complementary events, compound events, independent events and conditional probability to solve problems.

### Outcomes

#### VCMSP347

Describe the results of two- and three-step chance experiments, both with and without replacements, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence.

#### VCMSP348

Use the language of ‘if ....then, ‘given’, ‘of’, ‘knowing that’ to investigate conditional statements and identify common mistakes in interpreting such language.