1. ## probability question

A college delegation is made up of 6 basketball players, 5 volleyball players, 4 netball players, 3 badminton players and 2 tennis players. A subcommittee of four people is to be formed from these 20 athletes.

Find the probability that the subcommittee contains exactly one basketball player or exactly one volleyball player (or both).

My method:

let A be the event that committee has one basketball player
let B be event that committee has one volleyball player

so we want P (A U B)= P(A) + P(B) - P (A n B)

(n is intersect)

P(A) = 6C1 x 14C3/20C4
P(B) = 5C1 x 15C3/20C4
P(A n B) = 6C1 x 5C1 x9C2/20C4

my answer is 0.697 but the answer given is 0.414 can someone tell me where i went wrong? thanks~

2. ## Re: probability question

Originally Posted by thesocialnetwork
A college delegation is made up of 6 basketball players, 5 volleyball players, 4 netball players, 3 badminton players and 2 tennis players. A subcommittee of four people is to be formed from these 20 athletes.
Find the probability that the subcommittee contains exactly one basketball player or exactly one volleyball player (or both).
my answer is 0.697 but the answer given is 0.414 can someone tell me where i went wrong? thanks~
The wording is quite awkward. The following gives the suggested answer:
$\dfrac{\dbinom{6}{1}\dbinom{9}{3}+\dbinom{5}{1} \dbinom{9}{3}+\dbinom{6}{1}\dbinom{5}{1}\dbinom{9} {2}}{\dbinom{20}{4}}.$

Can you tell us why?

3. ## Re: probability question

yes, you take the 3 cases separately: (1) where there is exactly one basketball player and no volleyball player, (2) where there is exactly one volleyball player and no basketball player and (3) where there is exactly one basketball player and exactly one volleyball player, and then you divide the total number of outcomes in these 3 events by the total possible number of outcomes.

Technically though, isn't the method I first used correct? Where I add up P(A) and P(B) and subtract the intersect? Did I make a calculation error? I would've thought I answered the question properly.

4. ## Re: probability question

I think that what is happening is that your event A could include exactly one of each and so could your event B.

5. ## Re: probability question

Originally Posted by thesocialnetwork
Technically though, isn't the method I first used correct? Where I add up P(A) and P(B) and subtract the intersect? Did I make a calculation error? I would've thought I answered the question properly.
Your event A includes the case of exactly one baseball player and possibly three volleyball players. We don't allow that. We want exactly one baseball player, or exactly one volleyball player or exactly one of each. Those are exclusive or's.

6. ## Re: probability question

yes i see - the overlap is greater than i thought and my P (A n B) doesn't account for it. thank you very much! It's appreciated.