Year 9AC9M9P02

Probability with Venn Diagrams

Calculate relative frequencies from given or collected data to estimate probabilities of events involving “and”, inclusive “or” and exclusive “or”.

Read carefully. The details are important and often hidden in the words.

Method

Probability of an event

P(A) = \frac{n(A)}{n(S)}

How many ways it can happen, over how many outcomes there are.

Events that overlap

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

Take the overlap off, or you count it twice.

Mutually exclusive events

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

No overlap, so nothing to subtract.

Complementary events

P(A') = 1 - P(A)

Everything that is not $A$.

e.g.

\text{40 in a class. 7 are left-handed. 18 play soccer. 4 do both.}

Start with the overlap. 4 do both, so put 4 in the middle $n(S \cap L) = 4$
Take the overlap off each total. $7 - 4 = 3$ and $18 - 4 = 14$ $3 \text{ and } 14$
Everyone else goes outside. $40 - 3 - 4 - 14 = 19$ $19$
Write down the formula you need $P(S \cup L) = P(S) + P(L) - P(S \cap L)$
Substitute from the diagram $= \tfrac{18}{40} + \tfrac{7}{40} - \tfrac{4}{40} = \tfrac{21}{40}$
For right-handed AND soccer, read it straight off the diagram $P(R \cap S) = \tfrac{14}{40} = \tfrac{7}{20}$

Wrong

P(A \cup B) = P(A) + P(B) \text{ when the sets overlap}

Right

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

Adding straight off double-counts everyone in the middle. Only drop the last term when the events cannot both happen.

Wrong

\text{Putting the total in the overlap}

Right

\text{Fill the overlap first, then subtract}

If 18 play soccer and 4 of them are left-handed, only 14 go in the soccer-only part. The 18 already includes the 4.

Wrong

\text{Forgetting the people outside both circles}

Right

\text{All four regions must add to } n(S)

Always check the four numbers total the whole group. It catches most errors on the spot.

In a group of 30: 12 play tennis, 15 play netball, 5 play both.

\text{How many play tennis only?}

Answer

7

\text{How many play netball only?}

Answer

10

\text{How many play neither?}

Answer

8

P(\text{tennis})

Answer

\tfrac{12}{30} = \tfrac{2}{5}

P(\text{tennis and netball})

Answer

\tfrac{5}{30} = \tfrac{1}{6}

P(\text{tennis or netball})

Answer

\tfrac{22}{30} = \tfrac{11}{15}

P(\text{neither})

Answer

\tfrac{8}{30} = \tfrac{4}{15}

P(\text{not tennis})

Answer

\tfrac{18}{30} = \tfrac{3}{5}

P(\text{netball only})

Answer

\tfrac{10}{30} = \tfrac{1}{3}

\text{Check: do all four regions add to 30?}

Answer

7 + 5 + 10 + 8 = 30 \ \checkmark