Problem statement

Pólya is playing about with his urn again and he wants you to help him calculate some probabilities.

In this urn experiment Pólya has an urn which initially contains 1 red and 1 blue bead.

For every iteration, he reaches in and retrieves a bead, then inspects the colour and places the bead back in the urn.

He then flips a fair coin, if the coin lands heads he will insert a fair 6 sided die roll amount of the same coloured bead into the urn, if it lands tails he will remove half the number of the same colored bead from the urn (Using integer division - so if the number of beads of the selected colour is odd he will remove (c-1)/2 where c is the number of beads of that colour)

Given an integer n ≥ 0 and a decimal r > 0, give the probability to 2 decimal places that the ratio between the colours of beads after n iterations is greater than or equal to r in the shortest number of bytes.

An example set of iterations:

Let (x, y) define the urn such that it contains x red beads and y blue beads.

Iteration Urn Ratio
0 (1,1) 1
1 (5,1) 5 //Red bead retrieved, coin flip heads, die roll 4
2 (5,1) 5 //Blue bead retrieved, coin flip tails
3 (3,1) 3 //Red bead retrieved, coin flip tails
4 (3,4) 1.333... //Blue bead retrieved, coin flip heads, die roll 3

As can be seen the Ratio r is always ≥ 1 (so it's the greater of red or blue divided by the lesser)

Test cases:

Let F(n, r) define application of the function for n iterations and a ratio of r

F(0,5) = 0.00
F(1,2) = 0.50
F(1,3) = 0.42
F(5,5) = 0.28
F(10,4) = 0.31
F(40,6.25) = 0.14

This is code golf, so the shortest solution in bytes wins.

JavaScript (ES7),  145 139 135 132  129 bytes

Takes input as (r)(n). This is a naive solution that actually performs the entire simulation.

r=>g=(n,B=!(s=0),R=1,h=d=>++d<7?h(d,[0,d].map(b=>g(n,b?-~B>>1:B,b?R:-~R>>1)&g(n,B+b,R+d-b))):s/24**-~n)=>n--?h``:s+=B/R>=r|R/B>=r

Try it online!

Too slow for the last 2 test cases.