Year 10AC9M10P01

Tree Diagrams, Independence and Venn

Use the language of if-then, given, of, knowing that to describe and interpret situations involving conditional probability.

Two events, three tools. Which one to reach for depends on whether the events happen in stages or overlap.

Builds onYr 10 · Conditional Probability

Method

Independent: one does not affect the other

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

With replacement, or separate objects.

Dependent: the first changes the second

Without replacement. Update the totals.

Tree diagrams for stages

Multiply along, add across.

Venn diagrams for overlap

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

Subtract the overlap once.

Mutually exclusive means no overlap

P(A \cap B) = 0

Then the addition rule has nothing to subtract.

e.g.

\text{A bag holds 6 red and 4 blue. Two are drawn without replacement. Find } P(\text{one of each}).

Note the events are dependent. The bag changes. $\text{without replacement}$
There are two ways to get one of each. $RB \text{ or } BR$
Find the first path, multiplying along the branch. $\frac{6}{10} \times \frac{4}{9} = \frac{24}{90}$
Find the second path. $\frac{4}{10} \times \frac{6}{9} = \frac{24}{90}$
Add across the branches. $\frac{48}{90}$
Simplify. $= \frac{8}{15}$

Decide independent or dependent first. Then pick the tool.

\text{Two coins. } P(\text{two heads})?

Answer

\frac{1}{4}

\text{Same bag, with replacement. } P(\text{two red})?

Answer

\frac{6}{10} \times \frac{6}{10} = \frac{9}{25}

P(A) = 0{,}5,\ P(B) = 0{,}4,\ P(A \cap B) = 0{,}2. \text{ Find } P(A \cup B).

Answer

0{,}7

\text{Are A and B independent above?}

Answer

\text{Yes. } 0{,}5 \times 0{,}4 = 0{,}2 \quad\checkmark

\text{Two mutually exclusive events. Find } P(A \cap B).

Answer

0

\text{Why does without replacement change the second probability?}

Answer

\text{Both the total and the count have dropped.}