How many marbles were there originally

Question

A box Contains marbles.

Four of the marbles are there.

The rest of the marbles are yellow.

Antonia takes at random a marble from the box and does not replace it.

Sergio then takes at random a marble from the box.

The probability that Antonia and Sergio both take out a yellow marble is 0.7

Answer 1

Let A = Event that the First Marble is Yellow.

Let B = Event that the 2nd Marble is Yellow.

Let P(B|A) = probability that event B happens assuming that event A has happened.

0.7 = P(A) P(B|A)

P(A) = (N - 4) / N

P(B|A) = (N - 5) / (N-1)

We need to solve:

0.7 = ((N-4) / N) ( (N-5) / (N-1))

0.7 (N) (N-1) = (N-4) * (N-5)

(0.7) (N² - N ) = N² - 9N + 20

0.7N² - 0.7N = N² - 9N + 20

0 = 0.3N² - 8.3N + 20

0 = 3N² - 83N + 200

N = 2.667 or 25

Allowed values for N are positive integers great than 5.

N = 25, answer.

Answer 2

Assume that number of yellow marbles is x

So total marbles will be x + 4

P (yellow by Antonia) = x / (x+4)

P (yellow by Sergio) = (x-1)/(x+3)

It is (x-1) for yellow because Antonia doesn't replace it.

Total marbles will be one less after Antonia takes it

P (both yellow by Antonia & Sergio) = 0.7

x /(x+4) * (x-1)/(x+3)=7/10

(x(x-1))/((x+4) (x+3) = 7/10 cross multiple

10(x^2 - x) = 7 (x^2 +7x +12)

10x^2 - 10x = 7x^2 +49x + 84

3x^2 - 59x - 84 = 0

(x - 21) (3x + 4) = 0

(x - 21) =0 or (3x + 4) = 0

x = 21 or x = - 4/3 (N/A)

# yellow marbles = 21

Total marbles originally = 25

Proof :

21/25 * 20/24 = 0.7