Recall the probability of event $A$A occurring can be calculated as
$P\left(A\right)=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$P(A)=Number of favourable outcomesTotal number of outcomes
Combinations give us a new technique for counting the number of favourable and total outcomes.
Worked example
Example 1
A magazine editor is deciding which $3$3 stories to feature on the front cover of a magazine.
(a) If she has $5$5 excellent articles to choose from and decides to choose randomly, how many different selections are possible?
Think: Is order important and are there any restrictions? Order does not appear to be important and there are no restrictions apparent, so we can proceed with the combination formula.
Do: Using $^nC_r=\frac{n!}{r!\left(n-r\right)!}$nCr=n!r!(n−r)!, where $n$n is the total number of articles, $n=5$n=5 and $r$r is the number we are choosing, $r=3$r=3.
| $^5C_2$5C2 | $=$= | $\frac{5!}{3!2!}$5!3!2! |
| $=$= | $10$10 |
So there are $10$10 possible selections.
(b) If you were the writer of one of the stories, what is the probability that your article is chosen for the front page?
Think: We have the total number of outcomes, $10$10. We need to calculate the number of favourable outcomes and then we can calculate the probability.
Do: There are three articles picked for the front cover, favourable outcomes will have your article selected plus two out of the remaining four articles.
Hence,
| $\text{Number of favourable outcomes}$Number of favourable outcomes | $=$= | $^4C_2$4C2 |
| $=$= | $\frac{4!}{2!2!}$4!2!2! | |
| $=$= | $6$6 |
Thus the probability becomes,
| $P\left(\text{Article selected}\right)$P(Article selected) | $=$= | $\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Number of favourable outcomesTotal number of outcomes |
| $=$= | $\frac{6}{10}$610 | |
| $=$= | $\frac{3}{5}$35 or $0.6$0.6 |
Combinations and conditional probability
Recall that conditional probability involves conditions or restrictions are placed on the total number of outcomes, thus reducing the size of our sample space.
When considering whether a question is referring to a conditional probability, remember to look out for the following phrases:
- Given that ....
- If ......., what is the probability that .....
We can calculate the probability of event $A$A given event $B$B has occurred by using these formula. Remember that the condition defines the number on the denominator.
$P\left(A|B\right)=\frac{n\left(A\cap B\right)}{n\left(B\right)}$P(A|B)=n(A∩B)n(B) OR
$P(A|B)=\frac{P\left(A\cap B\right)}{P\left(B\right)}$P(A|B)=P(A∩B)P(B)
Let's look at some examples involving combinations.
Worked examples
Example 2
For lunch each day, Tom eats a sandwich, a sausage roll or sushi and he drinks a coffee, a tea or an orange juice.
(a) What is the probability that Tom eats a sausage roll for lunch?
Think: Are there any special conditions placed on this question? The answer is no, so we go ahead and calculate the number of favourable events and total number of outcomes using combinations. Notice there is an 'and' statement. So we will find the combinations of lunches and drinks separately and then multiply.
Do: Favourable outcomes are those where the sausage roll is selected for lunch:
| $\text{Total favourable outcomes}$Total favourable outcomes | $=$= | $^1C_1\times^3C_1$1C1×3C1 |
| $=$= | $1\times3$1×3 | |
| $=$= | $3$3 |
| $\text{Total Number of outcomes}$Total Number of outcomes | $=$= | $^3C_1\times^3C_1$3C1×3C1 |
| $=$= | $3\times3$3×3 | |
| $=$= | $9$9 |
Hence,
| $P\left(\text{Sausage Roll}\right)$P(Sausage Roll) | $=$= | $\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Number of favourable outcomesTotal number of outcomes |
| $=$= | $\frac{3}{9}$39 | |
| $=$= | $\frac{1}{3}$13 |
(b) Given that Tom drinks a coffee with his lunch, what is the probability he also ate a sandwich?
Think: Is there a condition placed on this question? Yes - we know Tom drinks a coffee, thus reducing the size of our sample space.
As we have not calculated any probabilities, let's calculate the probability directly with the numbers in the appropriate events:
$P\left(A|B\right)=\frac{n\left(A\cap B\right)}{n\left(B\right)}$P(A|B)=n(A∩B)n(B)
$P\left(\text{Sandwich}|\text{Coffee}\right)=\frac{n\left(S\cap C\right)}{n\left(C\right)}$P(Sandwich|Coffee)=n(S∩C)n(C)
Do: There is only one option that has a sandwich and a coffee. The options that have a coffee are number of different selections we have for a food item $^3C_1=3$3C1=3.
Hence,
| $P\left(\text{Sandwich}|\text{Coffee}\right)$P(Sandwich|Coffee) | $=$= | $\frac{n\left(S\cap C\right)}{n\left(C\right)}$n(S∩C)n(C) |
| $=$= | $\frac{1}{3}$13 |
Example 3
Elina is taking three books with her on holiday.
She has $4$4 Mathematics textbooks to choose from, $5$5 novels and $3$3 biographies.
(a) What is the probability she takes one of each?
| $\text{Number of favourable outcomes}$Number of favourable outcomes | $=$= | $\nCr{4}{1}\times\nCr{5}{1}\times\nCr{3}{1}$4C1×5C1×3C1 |
| $=$= | $60$60 |
| $\text{Total number of outcomes}$Total number of outcomes | $=$= | $\nCr{12}{3}$12C3 |
| $=$= | $220$220 |
Thus the probability becomes,
| $P\left(\text{One of each}\right)$P(One of each) | $=$= | $\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Number of favourable outcomesTotal number of outcomes |
| $=$= | $\frac{60}{220}$60220 | |
| $=$= | $\frac{3}{11}$311 |
(b) Given that she took only one Mathematics textbook, what is the probability she took at least one novel?
Think: Is there a condition placed on this question? Yes - we know that only one Mathematics textbook was taken, thus reducing the size of our sample space.
Using the conditional probability rule, we have:
$P\left(\text{At least 1 novel}|\text{Only 1 Maths}\right)=\frac{n\left(\text{At least 1 novel}\cap\text{Only 1 Maths}\right)}{n\left(\text{Only 1 Maths}\right)}$P(At least 1 novel|Only 1 Maths)=n(At least 1 novel∩Only 1 Maths)n(Only 1 Maths)
Do:
Options with only $1$1 Mathematics Textbook and at least $1$1 novel: $=\nCr{4}{1}\times\nCr{5}{1}\times\nCr{3}{1}+\nCr{4}{1}\times\nCr{5}{2}\times\nCr{3}{0}$=4C1×5C1×3C1+4C1×5C2×3C0
Options with only $1$1 Mathematics Textbook: $\nCr{4}{1}\times\nCr{8}{2}$4C1×8C2
Hence,
| $P\left(\text{At least 1 novel}|\text{Only 1 Maths}\right)$P(At least 1 novel|Only 1 Maths) | $=$= | $\frac{60+40}{112}$60+40112 |
| $=$= | $\frac{25}{28}$2528 |
Practice questions
QUESTION 1
A box contains 6 pens of different colours: red, green, blue, yellow, black and white. Two pens are drawn at random without replacement.
How many possible selections are there?
What is the probability of drawing the green and black pens?
QUESTION 2
A menu has three entrées ($E_1,E_2,E_3$E1,E2,E3), four mains ($M_1,M_2,M_3,M_4$M1,M2,M3,M4) and two desserts ($D_1,D_2$D1,D2). A meal is made up of one of each.
How many different meals are possible?
What is the probability of selecting $E_1$E1 , $M_3$M3 and $D_2$D2?
How many different meals are possible given that $E_1$E1 is the entrée?
QUESTION 3
$5$5 people are to be selected from a larger group of $10$10 candidates. If Amelia is among the candidates, what is the probability that she will be among those selected?
Question 4
A student is choosing two units to study at university: a language and a science unit. They have $4$4 languages and $7$7 science units to choose from.
If they choose one of each, what is the total number of combinations of choices?
If Italian is one of the languages they can choose from, what is the probability they choose Italian as their language?
French is one of the available languages. What is the probability they choose French as their language given that they choose Chemistry as their science unit?
Question 5
A tour guide is taking groups on holiday and wants to mix up the nationalities of the groups, but she does this at random. Each group has $6$6 people.
On this tour there are $10$10 Australians, $7$7 Americans and $12$12 Chinese people.
In one group, what is the probability that there are $2$2 people of each nationality?
Given that there are exactly $2$2 Chinese people in the group, what is the probability that there are at least $3$3 Americans?
What is the probability that there are no Australians in the group given that there are exactly $2$2 Americans?