This Exam P sample reference tests Geometric-Series Probability. After N red sectors, the blue area is 2/11+(9/11)(9/20)ᴺ. It remains above 0.20 at N=4 and falls below at N=5, choice C.
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=(209)i,BN=1−i=1∑NRi
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=112+119(209)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<0.20
Answer
Answer
Five red sectors are necessary and sufficient.
5(C)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AWith only three sectors, the remaining blue area exceeds even the nonqualifying four-sector value of 0.21537.
BFour sectors leave blue area 0.21537, which is still above the required 0.20 threshold.
DSix sectors satisfy the inequality, but five sectors already do so; six is not the minimum.
ESeven sectors also satisfy the inequality but are two sectors beyond the first qualifying count.
Original practice · fully worked
Original variant: panels covering a display
A display begins entirely white. Panel i covers a previously white area equal to (1/3)ⁱ of the display, and the panels do not overlap. Find the minimum number of panels needed to make the remaining white area less than 0.505.
A 2 panels
B 3 panels
C 4 panels
D 5 panels
E 6 panels
Variant answer in brief
The uncovered area after N panels is 1/2+(1/2)(1/3)ᴺ. It is 0.50617 after four panels and 0.50206 after five, so the minimum is five, choice D.
Setup
Setup
After any number of nonoverlapping panels, the uncovered area is one minus the finite geometric sum of the covered areas.
WN=1−i=1∑N(31)i
Model
Model
The finite-series formula gives a decreasing uncovered-area sequence with limiting value one half.
WN=21+21(31)N
Compute
Compute
Four panels leave 0.506173 uncovered, which is above 0.505. Five leave 0.502058, so five is the minimum.
W4=0.506173>0.505,W5=0.502058<0.505
Answer
Answer
Five panels first bring the uncovered area below the target.
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