I don't understand and my teacher won't explain!? (help me with probability please)

Question

Allie has a bag containing 3 white pom-poms and some black pom-poms. She shares that the probability of picking two white pom-poms from the bag without replacement is 6/20. How many black pom-poms are in the bag?

Answer 1

lets say there is x number of black pom-poms (X BLACK)

BAG ONLY CONTAINS 3 WHITE POM (3 WHITE)

TOTAL POMS in the bag = 3+X

NOW,

just think what the question is saying.

it tells that allie's probability to pick 2 white balls without replacing 1st ball that she withdraws is 6/20.

this means,

her possibility to pull 2 white out of total is 6/20.

Hence,

2 over 3+x is equal to 6/20.

Answer 2

Let's say there are x black pompoms and then the total number of pompoms is x+3.

Probability of picking the first white pompom is 3/(x+3), if we put it aside, now we have 2 white and x black in the bag, for a total of (x+2). That means the probability of picking the second white is 2/(x+2).

That makes the probability of picking two white without replacement =(3/(x+3))*(2/(x+2))=6/((x+2)(x+3))=6/20.

Therefore (x+2)(x+3)=20, 4*5=20 x+2=4 and x+3=5. Then you can see that x must be 2.

Or you can foil and solve like this (x+2)(x+3)=20 x^2+5x+6=20 x^2+5x-14=0 if you use quadratic formula or factoring, you will get x=-7 or x=2, -7 is not acceptible, because you can't have negative number of pom-poms, so the answer is 2.

Hope this helps.