Probability of 3 specific players coming top 3 in a game of 7

[Mos5180d]

Ive definitely over thought this!

jenny, Shane and bill are playing a game with seven players (including them)

what is the probability that they all come within the top 3?

I’ve been playing around with permutations and combinations and all sorts but I’m actually thinking it’s as simple as

3/7 x 2/6 x 1/5 = 1/35

 

[Dr.Peterson]

Yes, that's correct. (I initially thought it wasn't, because you didn't state your reasons and I saw it the wrong way.) The probability that one of the three gets first place is 3/7; given that, the probability that another of them gets second place is 2/6; and so on. (This assumes, by the way, that there are no ties.)

But you can also use either permutations or combinations.

Permutations: How many ways can place numbers be assigned to everyone, so that these three get 1, 2, 3? How many ways can place numbers be assigned, total? Divide.

Combinations: How many ways can these three be chosen? (1!) How many ways can any three be chosen? Divide.

It will be good experience to try doing it all three ways, and then check that you get the same answers. If you don't, write back showing your work for each method, and we can comment.

 

[pka]

Ive definitely over thought this!
Jenny, Shane and bill are playing a game with seven players (including them)
what is the probability that they all come within the top 3?
I’ve been playing around with permutations and combinations and all sorts but I’m actually thinking it’s as simple as
3/7 x 2/6 x 1/5 = 1/35

There are \(\displaystyle 7!\) ways for the seven people can finish the game.
The three can be the top tree is \(\displaystyle 3!\) ways and there are \(\displaystyle 4!\) for the others.
\(\displaystyle \dfrac{(3!)(4!)}{7!}=\dfrac{1}{35}\)
So good for you being correct.

 

[JeffM]

You seem to be assuming that all the players have an equal probability of winning. That is a very unlikely assumption with respect to most games. I doubt that my probability of reaching the final four in the 2020 French Open equals that of Nadal.