Posted by Candice on .
A box contains 20 balls, of which 5 are red, 5 are blue, 5 are white and 5 are green. Suppose a sample of 5 balls are chosen without replacement.
a) find the probability that the sample contains balls of all the same colour
(answer is 4/ 20C5, but why isnt it 5/20C5??)
b) find the probability that the sample contains three balls of one colour and two balls of another colour.
The teacher wrote
5C3 x 5C2 x 4P2
Could you please explain why this is?
a) the ans. is 4/20C5 because there are 4 types of balls and 5 is the magnitude of a specific colour.
b)5C3 gives value of 3 balls of 1 colour;
5C2= value of 2 balls of another colour;
4P2= probality of 2 colours from 4 colours ;
20C5=combination of total balls of different colour.
could be all red, all blue , all white or all green
prob(all red) = (5/20)(4/19)(3/18)(2/17)(1/16) = /15504
the same calculation for the others
so the prob(all same colour) = 4(1/15504) = 1/3876
which is the same as 4/C(20,5)
Why isn't it 5/C(20,5) ??
Why would it? There are only 4 colours
3 colours of one kind, 2 of the others :
RRRBB RRRWW RRRGG
There would be 12 of those
let's look at one such case
prob = C(5,3) x C(5,2)/C(20,5) = 10/15504
but there are 12 cases with the same colour triple, and another colour double
so prob = 12 x C(5,3) x C(5,2)/C(20,5)
which is the same as your teacher's answer