A teacher wants to randomly select 2 students from a group of 5 students. The names of the students are John, Anna, Maria, Steve and Sean.
(a) Define the set of basic outcomes for this problem. What is the probability of each outcome?
(b) Let A = “the name of at least one of the selected students consists of four letters”, B = “at least one of the selected students is a boy”. Find the probabilities of these events.

### (a) Defining the set of basic outcomes and calculating the probability of each outcome

The teacher wants to randomly select 2 students from a group of 5 students (John, Anna, Maria, Steve, and Sean). To find the set of basic outcomes, we need to list all the possible pairs of students.

The number of ways to choose 2 students from 5 can be calculated using combinations:
\[
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
\]
So, there are 10 possible outcomes. The set of basic outcomes (which are the possible pairs of students) is:
\[
S = \{ (John, Anna), (John, Maria), (John, Steve), (John, Sean), (Anna, Maria), (Anna, Steve), (Anna, Sean), (Maria, Steve), (Maria, Sean), (Steve, Sean) \}
\]

Each of these outcomes is equally likely, so the probability of any specific pair being selected is:
\[
P(\text{each outcome}) = \frac{1}{10}
\]

### (b) Finding the probabilities of events A and B

#### Event A: "At least one of the selected students has a name consisting of four letters"
We need to identify which students have names with four letters. These students are:
- John
- Anna
- Sean

Thus, the favorable outcomes for event A are the pairs that include at least one of John, Anna, or Sean. The favorable pairs are:
\[
A = \{ (John, Anna), (John, Maria), (John, Steve), (John, Sean), (Anna, Maria), (Anna, Steve), (Anna, Sean), (Maria, Sean), (Steve, Sean) \}
\]
There are 9 such outcomes, so the probability of event A is:
\[
P(A) = \frac{9}{10}
\]

#### Event B: "At least one of the selected students is a boy"
The boys in the group are John, Steve, and Sean. Therefore, we need to find the pairs that include at least one of these students. The favorable pairs for event B are:
\[
B = \{ (John, Anna), (John, Maria), (John, Steve), (John, Sean), (Anna, Steve), (Anna, Sean), (Maria, Steve), (Maria, Sean), (Steve, Sean) \}
\]
There are 9 such outcomes, so the probability of event B is:
\[
P(B) = \frac{9}{10}
\]

### Questions for further exploration:
1. How would the probabilities change if the group size was increased to 6 students?
2. What is the probability of selecting exactly one boy from the group?
3. How would you calculate the probability of selecting two students with different numbers of letters in their names?
4. What is the probability of selecting a pair of students where both are girls?
5. Can we compute the probability of event A ∩ B (both events A and B happening)?

### Tip:  
When calculating probabilities involving combinations, always check if the events are mutually exclusive or can overlap to avoid errors in counting.

A teacher wants to randomly select 2 students from a group of 5 students. The names of the students are John, Anna, Maria, Steve, and Sean.
(a) Define the set of basic outcomes for this problem. What is the probability of each outcome?
(b) Let A = 'the name of at least one of the selected students consists of four letters', B = 'at least one of the selected students is a boy'. Find the probabilities of these events.

Learn how to solve a combinatorics and probability problem involving selecting 2 students from a group of 5. The problem covers the calculation of basic outcomes, as well as the probabilities of selecting students based on name length and gender. Suitable for students in Grades 9-12.

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