Question

You have a bag of 20 marbles. Twelve of the marbles are green and eight of the marbles are yellow. You draw three marbles out of the bag without replacement. What is the probability that all three marbles will be yellow? Is this an example of a binomial experiment?

Answer 1

The combined probability of picking three yellow marbles
`= ((14)/(285) )`

It is NOT a binomial experiment.

Explanation:

Here, the total number of marbles in the bag = 20

The total number of green marbles = 12

The total number of yellow marbles = 8

Now, let E : Event of picking a picking a yellow marble

`P(E) = \frac{\textrm{THe total number of yellow marbles}}{\textrm{The total marbles in the bag}}`

So, the probability of picking FIRST yellow marble =
`(8)/(20) = ((2)/(5))`

Now,as the marbles is again picked WITHOUT REPLACEMENT:

So, total marbles in the bag now = 20 -1 = 19 and yellow marbles = 8 -1 = 7

P(Picking second yellow marble) =
`((7)/(19) )`

P(Picking Third yellow marble) =
`((6)/(18) ) = ((1)/(3) )`

Now the combined probability of picking three yellow marbles

=
`((2)/(5) ) * ((7)/(19) ) * ((1)/(3) ) = ((14)/(285) )`

Also, the binomial experiment =
`^n C_x(p)^x(q)^{n-x`

Solving for n = 20 and x = 3 , p = 2/5 and q = 3/5 we get:

P =
`^(20)C_3 ((2)/(5))^3((3)/(5) )^(17) = 1140 * (8)/(125) *(3^(17))/(5^(17))` ≠
`((14)/(285))`

Hence, it is NOT a binomial experiment.