Independent solution

How to solve this Geometric-Series Probability question

Setup

Setup

Let each red sector's area be the stated geometric term. Since the sectors do not overlap, the remaining blue area is one minus the finite sum of all red-sector areas.

Ri=(920)i,BN=1−∑i=1NRiR_i=\left(\frac9{20}\right)^i,\quad B_N=1-\sum_{i=1}^N R_i

Model

Model

Summing the geometric series gives a decreasing expression for the blue area after any number of sectors. The minimum is established by checking the two adjacent candidate counts.

BN=211+911(920)NB_N=\frac2{11}+\frac9{11}\left(\frac9{20}\right)^N

Compute

Compute

After four sectors the blue area is 0.21537, still above 0.20. After five sectors it is 0.19692, so five is the first qualifying count.

B4=0.21537>0.20,B5=0.19692<0.20B_4=0.21537>0.20,\quad B_5=0.19692<0.20

Answer

Answer

Five red sectors are necessary and sufficient.

5(C)\boxed{5\quad\text{(C)}}