Balls in bags

Let's let the event be $E.$ Then Bayes' theorem says $P(RR|E) = \frac{P(E|RR)P(RR)}{P(E)}.$ Now $P(E|RR) = 1,$ and $P(RR) = 1/4,$ so the only difficult calculation is $P(E).$

But $P(E) = P(E|GG)P(GG) + P(E|GR)P(GR) + P(E|RR)P(RR),$ and $\begin{aligned} P(E|GG) &= \frac13 \cdot \frac13 \cdot \frac1{10} \\ P(GG) &= \frac14 \\ P(E|GR) &= \frac23 \cdot \frac23 \cdot \frac25 \\ P(GR) &= \frac12 \\ P(E|RR) &= 1 \\ P(RR) &= \frac14 \end{aligned}$ so we get $P(RR|E) = \frac{1/4}{1/360 + 4/45 + 1/4} = \frac{30}{41},$ so the answer is $\fbox{71}.$

Let's go through all cases: Both Green: Adding N Red balls, taking N out at random and them all being Red, has probability $\frac{2}{(N+2)(N+1)}$ . (Probability of picking 2 Green balls to be left).

One Red, One Green (twice as likely): Adding N Red balls, taking N out at random and them all being Red, has probability $\frac{2}{(N+2)}$ . (Probability of picking 1 Green balls with 2 picks to be left).

Both Red: Adding N Red balls, taking N out at random and them all being Red, has probability $1$ .

Ratios are:

$2 : 2*2(N+1) : (N+2)(N+1)$

Doing this multiple times, with $N = a,b,c$ gives the ratio:

$2^3 : 2 * 2^3(a+1)(b+1)(c+1) : (a+2)(a+1)(b+2)(b+1)(c+2)(c+1)$

Setting $a=b=1, c=3$ :

$2^3 : 2 * 2^3 * 2^4 : 2^4 * 45 == 1 : 32 : 90$

So the odds that both balls were Red are: $\frac{90}{123} = \frac{30}{41}$

You start out with rr, gg or gr with respective probabilities 1/4, 1/4 and 2/4.

After adding a red ball and randomly retrieving a ball, what are the chances you retrieve a red ball?

P(r|gg)=1/3

P(r|rr)=1

P(r|gr)=2/3

After that, the situation is back to what it was before you did this. So doing this three times in a row gives these chances:

P(3 times r|gg)=1/27

P(3 times r|rr)=1

P(3 times r|gr)=8/27

P(3 times r AND gg) =1/4*1/27=1/108

P(3 times r AND rr) =1/4*1=1/4=27/108

P(3 times r AND gr) =1/2*8/27=16/108

Therfore p(3 times r) =(1+27+16)/108

P(3 times r AND rr) = P(3 times r) *P(rr|3 times r)

27/108=44/108*P(rr|3 times r)

P(rr|3 times r)=27/44

The answer is 27+44=71.