Probability of getting a certain group of students when choosing three at random out of 25

Question

A teacher randomly chooses a group of three students from her class of 25 students. Find:

a) Probability that friends Suri, Lily and Violeta are chosen for the group?

b) If he assign roles to the students as she chooses them. The first chosen will be the researcher, the second the writer and the third the illustrator. What is the probability that Suri will be the researcher, Lily the writer and Violeta the illustrator?

Progress

I tried question a):

Let A represent Suri , Lily and Violeta are chosen for the group. Let O represent all possible selections. The number of ways that Suri , Lily and Violeta can be chosen is n(A) = 3P3 = 3!. The number of ways that the 25 people can be chosen for the 3 positions is 25P3 $$n(O) = 25P3 = 13,800$$ Now determine the probability. $$P(A) = \frac{n(A)}{n(O)} = \frac{6}{13800} = \frac{1}{2300}$$ The probability Suri , Lily and Violeta will be is $\frac{1}{2300}$.

But I don't know if it's correct.

Answer 1

You got the right answer, but lets think of it from more of a conditional probability standpoint

So the probability of the first student chosen being one of them is just then number of them divided by the total number, i.e. $\frac{3}{25}$. Now when looking at the second one, we already know that one of the 3 students has been chosen and one of the 25 students has been chosen so we have $P(2=Lily, Violeta | 1=Suri) = \frac{2}{24}$. Then the third works the same way to be $\frac{1}{23}$. If you multiply these together you get $\frac{1}{2300}$ as you found. I'll try to make this section a little bit more clear:

So when we are picking the first student, we can pick either Suri, Lily, or Violeta. This gives us 3 possible choices. Our total sample size for this event (let's say event $A$) is $P(A)=\frac{3}{25}$. Ok, now we've already chosen one student (either Suri, Lily, or Violeta) so the subset that we can choose from now is only of size 2 and our total population is now only 24. Thus the probability of this event occurring (let's call it $B$) is $P(B|A)=\frac{2}{24}$. Now for the third part we have already selected two of the chosen students so our subset is now of size 1 and our population is size 23. Thus the probability of this event (let's call it $C$) is $P(C|A,B)=\frac{1}{23}$. So the overall probability of all three events happening is

$$ P(A)\cdot P(B|A) \cdot P(C|A,B)=\frac{3}{25} \cdot \frac{2}{24} \cdot \frac{1}{23}=\frac{1}{2300} $$

So for part (b) instead of having a subset of three that we can choose from for each job, we can only choose one student. So when we choose the first student, our total sample size is 25 and we must choose Suri (1 individual student). Let's call the event that we choose Suri to be the researcher $A$, we have $P(A)=\frac{1}{25}$. Now when we choose the writer we have already chosen a researcher so our sample size is now 24. We still have to pick one singular student though, Lily. So let's call this event $B$, and we have that $P(B|A)=\frac{1}{24}$. Now when we choose the illustrator we have already chosen 2 students so the population is now 23, thus we have $P(C|A,B)=\frac{1}{23}$. The intersection of these events gives us the total probability:

$$ P(A)\cdot P(B|A) \cdot P(C|A,B)=\frac{1}{25} \cdot \frac{1}{24} \cdot \frac{1}{23}=\frac{1}{13800} $$

Answer 2

I tried question a) is Let A represent Suri , Lily and Violeta are chosen for the group.Let O represent all possible selections.The number of ways that Suri , Lily and Violeta can be chosen is n(A) = 3P3 = 3!. The number of ways that the 25 people can be chosen for the 3 positions is 25P3
n(O) = 25P3 = 13,800
Now determine the probability.
P(A) = n(A)/n(O) = 6/13800 = 1/2300
The probability Suri , Lily and Violeta will be is 1/2300.

Answer 3

a) Probability that friends Suri, Lily and Violeta are chosen for the group?

There's one way to select the students out of all the ways to select 3 of 25 students (order doesn't matter).

$$\begin{align}\frac 1 {^{25}C_3} & = \frac{3!22!}{25!} \\ & = \frac{1\times 2\times 3}{23\times 24\times 25} \\ & = \frac{1}{2300}\end{align}$$

b) If he assign roles to the students as she chooses them. The first chosen will be the researcher, the second the writer and the third the illustrator. What is the probability that Suri will be the researcher, Lily the writer and Violeta the illustrator?

That depends on wether that's asking for the conditional probability that they will be assigned those roles given that they've been selected, or for the unconditional probability that they will be selected and assigned those roles.

$$\text{The conditional probability is}: \frac{1}{3!} = \frac 1 6$$

$$\text{The unconditional probability is}: \frac{1}{^{25}P_3} = \frac{1}{3!}\times\frac{3!22!}{25!}=\frac{22!}{25!} = \frac 1 {13800}$$