# Thread: A box contains 6 tennis balls

1. ## A box contains 6 tennis balls

A box contains 6 tennis balls. Peter picks two balls at random from the box, plays with them, and returns them to the box. Paul then picks two more balls at random from the box (they can be the same or different from Peter's balls), plays with them, and returns them to the box. Finally, Mary picks two more balls at random and plays with them. What is the probability that every ball picked was played with exactly once?

2. ## Re: A box contains 6 tennis balls

Hi,

do you have the solution?

Anyways, this was my approach (IDK if it is right), but I tried:

So we know peter picks two balls at random from the box and plays with them and returns them, which means permutations, or w/ replacement.
Mary is combinations, or w/o replacement

6 P 2 + (4 P 2)(2 P 0) + (2 C 2)(4 C 0) = 30+12+1= 43/ 75...

Again this is probably not right.. but I am trying to think how we know that every ball picked was played with at least once. We know that Peter has to play with 2 unique balls, same with Paul, and same with Mary. So for 6 P 2 my thinking was, out of 6 balls, we play with 2 and replace them. Then for Paul: given that we already played with 2 unique balls (2 P 0), what is left is 4 P 2, and then for Mary, given that we have played with 4 C 0 balls, what is left is 2 C 2. And for the probability space I just did (6 P 2 )^2+ 6 C 2

3. ## Re: A box contains 6 tennis balls

$$\dfrac{\dbinom 6 2 \dbinom 4 2 \dbinom 2 2}{\dbinom 6 2 \dbinom 6 2 \dbinom 6 2}=\dfrac{2}{75}$$

The idea is, the first person can pick any two balls. So, out of 6, choose 2. The second person cannot play with either of those two balls. That leaves 4, choose 2. The last person cannot choose any of the 4 already chosen. That leaves 2 to choose from 2.

The total number of ways to make those choices is 6 choose 2 three times (no restrictions for anyone)

4. ## Re: A box contains 6 tennis balls

Hi 'math951', thank you give me advise.
I think 'SlipEternal' reply is right. Every ball picked was played with exactly once probability is: 2/75

5. ## Re: A box contains 6 tennis balls

Originally Posted by Amanda2018
A box contains 6 balls.
Peter picks 2 balls at random from the box, and returns them to the box.
Paul then picks 2 balls at random from the box, and returns them to the box.
Mary then picks 2 balls at random.
What is the probability that every ball was picked?
Number the balls from 1 to 6.

Peter picks 5 and 6.

Paul must pick 2 numbered from 1 to 4 (from the 6 balls).

Mary must pick the 2 balls not picked yet.

Works out to 2/75 (as per Slip).

6. ## Re: A box contains 6 tennis balls

Yes... so simple, but I oversimplified it once again..

7. ## Re: A box contains 6 tennis balls

Given that there is a 1/6 chance of choosing a ball, I should of known, picking exactly 6 balls and each ball only being picked once should be < than 1/6.

8. ## Re: A box contains 6 tennis balls

Go stand in the corner for 1/6 * 60 minutes!!