Teams
Two brothers belong to a club with 10 members. Four are selected for a match. Find the probability that both brothers are selected.
Problem
A chess club has $10$ members, $4$ women and $6$ men, of whom two are brothers.
A team of $4$ members is selected to play a match. Suppose that each member has an equal chance of being selected what is the probability that both brothers are in the team and what is the probability that neither brother is in the team?
What is the probability that there are at least two women in the team and what is the probability that there are at least two men in the team?
Getting Started
How many ways can you choose a team of $4$ from $10$?
Student Solutions
Phil sent us this solution:
There are $^{10}C_4=210$ possible chess teams.
- To find the number of teams containing both brothers, we need the number of ways of choosing the other $2$ team members from the remaining $8$ people, which is $^8C_2=28$. So the probability is $28/210=2/15$.
- To find the number of teams containing neither brother, we need the number of ways of choosing all $4$ team members from the remaining $8$ people, which is $^8C_4=70$. So the probability is $70/210=1/3$.
- To find the number of teams containing at least $2$ women, we can take $210$ - the number of teams containing $0$ women - the number of teams containing $1$ woman. There are $^6C_4=15$ teams with $0$ women, and $4 \times$ $^6C_3=80$ with $1$ woman, so the required probability is $1-\frac{15}{210}-\frac{80}{210}=\frac{ 115}{210}=\frac{23}{42}$.
- Similar to the last one. There is only $1$ team containing $0$ men. There are $4\times$ $^6C_1=24$ teams containing $1$ man, so the probability we want is $1-\frac{1}{210}-\frac{24}{210}=\frac{185}{210}=\frac{37}{42}$.
Teachers' Resources
Why do this problem?
This problem gives students the opportunity to think about probability in "without replacement" situations and in particular, how you can deal with a range of constraints. Links can be made with the concept of combinations which may be familiar in the context of binomial coefficients.
Possible approach
Students may be nudged in the direction of using combinations from the outset by asking them first to consider how many ways there are to choose a team of 4 from 10 people (as in the Getting Started section). Alternatively, they could simply treat each choice of team member as an independent event. If this approach is taken, they may need reminding to think of all of the possible orders in which the events can occur, for example all of the possible orders of brothers and non-brothers being selected. They may then need convincing that, although the association of numerators with denominators may be different for different orders, the overall product is the same. It is worth considering whether you would like to encourage one approach over the other, allow students to develop their own strategies and stick with them provided they are valid, or whether finding different approaches and comparing them may be a goal of the activity.