Combined Probability

Spec reference: Probability
Key idea: Use tree diagrams and Venn diagrams to calculate probabilities of combined events.

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Independent events

Two events are independent if one does not affect the other.

$$P(A\text{ and }B)=P(A)\times P(B)$$

Example

A coin is flipped and a die is rolled. Find $P(\text{heads and 6})$.

$$P=\frac{1}{2}\times \frac{1}{6}=\boxed{{\displaystyle \frac{1}{12}}}$$

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Tree diagrams

Tree diagrams show all possible outcomes of combined events.

• Multiply along branches to find probability of a path
• Add probabilities of different paths for the same outcome

Example

A bag has 4 red and 6 blue balls. Two balls are drawn with replacement.

$P(\text{both red})=\frac{4}{10}\times \frac{4}{10}=\frac{16}{100}=\boxed{{\displaystyle 0.16}}$

$P(\text{one of each})=\frac{4}{10}\times \frac{6}{10}+\frac{6}{10}\times \frac{4}{10}=0.24+0.24=\boxed{{\displaystyle 0.48}}$

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Without replacement

When items are not replaced, the probabilities change for the second pick.

Example

A bag has 4 red and 6 blue. Two drawn without replacement.

$P(\text{both red})=\frac{4}{10}\times \frac{3}{9}=\frac{12}{90}=\boxed{{\displaystyle \frac{2}{15}}}$

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Conditional probability

$$P(A|B)=\frac{P(A\text{ and }B)}{P(B)}$$

$P(A|B)$ means "probability of $A$ given $B$ has happened."

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Venn diagrams

Two overlapping circles $A$ and $B$ within a rectangle (universal set $\xi$).

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Notation	Meaning
$A\cup B$	$A$ or $B$ (union)
$A\cap B$	$A$ and $B$ (intersection)
$A^{\prime }$	Not $A$ (complement)

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Exam tips

Watch out for

• With replacement: probabilities stay the same on all branches
• Without replacement: update the denominator (and numerator if needed) after each pick
• Venn diagram: the intersection is counted ONCE, not twice

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Test Yourself

Question 1 of 5

P(A) = 0.4, P(B) = 0.3, independent. P(A and B)?