Coaching Centre
Compound events, Venn diagrams and two-way tables
Events using and, or and not, mutually exclusive events, and organising outcomes in Venn diagrams and two-way tables to find probabilities.
And, or, not
Venn diagrams
In a class of $30$, $18$ play sport ($S$), $12$ play music ($M$) and $5$ do both. In a Venn diagram, the overlap is $5$, "sport only" is $18 - 5 = 13$, "music only" is $12 - 5 = 7$, and "neither" is $30 - 13 - 5 - 7 = 5$.
Using the class above, find $P(S \text{ or } M)$ and $P(\text{music only})$.
Two-way tables
A two-way table shows the same information in rows and columns, with totals.
| $M$ | not $M$ | Total | |
|---|---|---|---|
| $S$ | 5 | 13 | 18 |
| not $S$ | 7 | 5 | 12 |
| Total | 12 | 18 | 30 |
Class of $30$: $18$ sport, $12$ music, $5$ both. Find $P(\text{sport or music})$.
$\tfrac{25}{30}$.
Same class. Find $P(\text{music only})$.
$12 - 5 = 7$.
Same class. Find $P(\text{neither})$.
$\tfrac{5}{30}$.
Same class. Find $P(\text{both})$.
$\tfrac{5}{30}$.
Rolling a $2$ and rolling an odd number on one die are:
$2$ is not odd, so they cannot both happen.
A die is rolled. Find $P(\text{even or a } 3)$.
$\{2, 4, 6, 3\}$: $\tfrac{4}{6}$.
A die is rolled. Find $P(\text{even or greater than } 4)$.
$\{2, 4, 5, 6\}$ is $4$ outcomes: $\tfrac{4}{6}$.
In a group of $40$, $25$ like tea, $20$ like coffee and $5$ like neither. How many like both?
$40 - 5 = 35$ like at least one. $25 + 20 - 35 = 10$ are counted twice, so $10$ like both.
New numbers every time. Keep going until you get five in a row.