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Level 10

8.05 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
\displaystyle P(A)\displaystyle =\displaystyle 0.2 + 0.3Substitute P(A \cap B)=0.2,\,P(A \cap B')=0.3
\displaystyle =\displaystyle 0.5Evaluate the addition
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
\displaystyle P(B)\displaystyle =\displaystyle \dfrac{0.2}{0.5}Substitute P(A \cap B)=0.2 ,\, P(A)=0.5
\displaystyle =\displaystyle 0.4Evaluate the division
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
\displaystyle P(A \cup B')\displaystyle =\displaystyle P(A)+P(B')-P(A\cap B')Use the formula
\displaystyle P(A \cup B')\displaystyle =\displaystyle 0.5 + (1-0.4)-0.3Substitute the probabilities
\displaystyle =\displaystyle 0.8Evaluate

Example 2

Consider the following probability Venn Diagram:

A Venn diagram showing two intersecting circles labeled A and B. Ask your teacher for more information.

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
\displaystyle P(A|B)\displaystyle =\displaystyle \dfrac{0.2}{0.2+0.1}Substitute the probabilities from the Venn diagram
\displaystyle =\displaystyle \dfrac{2}{3}Evaluate
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.

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