10.05 Solving problems with probability

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 + 0.3	Substitute P(A \cap B)=0.2,\,P(A \cap B')=0.3
	\displaystyle =	\displaystyle 0.5	Evaluate the addition

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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 0.4	Evaluate the division

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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)-0.3	Substitute the probabilities
	\displaystyle =	\displaystyle 0.8	Evaluate

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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

\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

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