Probability!

Question

Hi Mike

I came across the below qs... i am unable to understand the difference between them please help me understand these. I have read all explanations available everywhere.

3) From a group of M employees, N will be selected, at random, to sit in a line of N chairs. There are absolutely no restrictions, either in the selection process nor in the order of seating — both are entirely random. What is the probability that the employee Andrew is seated somewhere to the right of employee Georgia?
Statement #1: N = 15
Statement #2: N = M

4) From a group of J employees, K will be selected, at random, to sit in a line of K chairs. There are absolutely no restrictions, either in the selection process nor in the order of seating — both are entirely random. What is the probability that the employee Lisa is seated exactly next to employee Phillip?
Statement #1: K = 15
Statement #2: K = J

I am unable to understand what the difference is... y the answers are different.

KevinRocci · Answer

Howdy! I think I might be able to help a little bit with these two problems.

The major difference between the two questions is in the question at the very end, specifically, in two short phrases: "Andrew is seated somewhere to the right of employee Georgia?" vs. "Lisa is seated exactly next to employee Phillip" I won't work through solving these problems, but let me try to explain the difference between this wording. When we have a question that says somewhere to the right of, this means that Andrew and Georgia could be seated next to each other or apart. There are going to be a lot more options for where they could sit. For example, just to make things easy to understand, let's say that there are 6 chairs to sit in. Here are all of the possible seating arrangements of Andrew (A) and Georgia (G) that meets the requirement of Georgia sitting to the right of Andrew. A G _ _ _ _ A _ G _ _ _ A _ _ G _ _ A _ _ _ G _ A _ _ _ _ G and it continues _ A G _ _ _ _ A _ G _ _ _ A _ _ G _ _ A _ _ _ G and it continues _ _ A G _ _ _ _ A _ G _ _ _ A _ _ G and _ _ _ A G _ _ _ _ A _ G finally _ _ _ _ A G That's a lot of options, right?

Now let's look at the other wording. "Lisa is seated exactly next to employee Phillip" would mean that Lisa and Philip are always next to each other. So Lisa (L) and Philip (P) are never apart. The possible arrangements for how they will sit in 6 chairs is less: L P _ _ _ _ _ L P _ _ _ _ _ L P _ _ _ _ _ L P _ _ _ _ _ L P and P L _ _ _ _ _ P L _ _ _ _ _ P L _ _ _ _ _ P L _ _ _ _ _ P L Does this make sense? So the wording actually is different enough to give us very different possibilities. Can you understand why the answers are different now? If not, show me your work and I can help you make sense of it.

Cheers!

NGGMAT · Answer

Hey Kevin

It is quiet clear now... i just had one major confusion...as to why knowing that M=N alone... without any number is sufficient for the first qs?
I tried using 5 chairs..

so the total number of outcomes = 5!
total number of favorable outcomes:

A G _ _ _
 
= 4! it will be 4! right??? as order matters.

I get stuck here. as to y we dont need a number and if i had to calculate how would i do it in PS?

In Qs 2, I am able to calculate it as a PS qs... but dont understand why here we need to know the value of K/J (K=J=15)... how does knowing the value matter?

KevinRocci · Answer

Happy to help!

So I realize now that I mixed up Andrew and Georgia in the seating arrangement. A should be the right of G!

The reason that M = N is sufficient to answer the question is because each employee with have a seat. No one will be left out. So whatever the total seats, we end up with everyone seated. That means we can calculate the total number of possibilities. Also we know that if Andrew is seated to the right of Georgia, it will give us a certain number of arrangements. Now think about this. There are an equal amount of ways to arrange everyone so that Georgia is seated to the right of Andrew. If we were to add both of those possibilities together we would have all possible arrangements. So we can see that the probability of Andrew sitting to the right of Georgia is half of the possible arrangements. Does that make sense? Let's use some numbers to make it easier.

Let's say there are 5 chairs and 5 people as you indicated. You are correct about the total possible outcomes—5! 5! = 120 Now how many outcomes are there where Andrew is to the right of Georgia? G A _ _ _ is one possibility = 3! = 6 You made a mistake by thinking that this is 4!. Think about it, though, once those two are seated, then you are thinking about the seating only 3 people—not 4. The rest of the possibilities are as follows: G _ A _ _ G _ _ A _ G _ _ _ A _ G A _ _ _ G _ A _ _ G _ _ A _ _ G A _ _ _ G _ A _ _ _ G A So really we can think of this as 5 options choose 3. Thus, 5C3 5! / 3! (5-3)! 5! / 3! 2! 5 * 4 *3 *2 *1 / 3 * 2 * 1 * 2 * 1 5 * 4 / 2 * 1 20 / 2 10 But we are not done. We only know the different ways that Georgia and Andrew can be seated at this point. We now need to account for the other people in these combinations. So we take the different ways that we can organize Andrew and Georgia and multiply by the number of ways that we can organize the rest of the people (3! = 6) So we have 6 * 10 = 60 The probability of Andrew sitting to the right of Georgia is thus: 60/ 120 or 1/2 Does that make sense? I hope that I've been able to help a little.

Cheers

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