Psurvive=1−Pcapture
The 1st three flies completed the quota (the 1st three flies were captured)
Q1=(1/2)3=1/8
The 4th fly completed the quota (only 2 were captured from the 1st three flies)
Q2=(1/2)2(1/2)1⋅(3C2)×(1/2)=3/16
The 5th fly completed the quota (only 2 were captured from the 1st four flies)
Q2=(1/2)2(1/2)2⋅(4C2)×(1/2)=3/16
Qquota = Probability that the spider made its quota of three flies before the 6th fly made the attempt to pass:
Qquota=Q1+Q2+Q3=1/8+3/16+3/16
Qquota=1/2 ← this is the probability that the spider will not attack the 6th fly
Pattack = Probability that the spider will attack the 6th fly
Pattack=1−Qquota=1−1/2
Pattack=1/2
Pcapture = Probability that the 6th fly will get caught by the spider
Pcapture=Pattack×Pcatch a fly=1/2×1/2
Pcapture=1/4
Hence,
Psurvive=1−1/4
Psurvive=3/4 ← answer