$P_\text{survive} = 1 - P_\text{capture}$
The 1st three flies completed the quota (the 1st three flies were captured)
$Q_1 = (1/2)^3 = 1/8$
The 4th fly completed the quota (only 2 were captured from the 1st three flies)
$Q_2 = (1/2)^2 (1/2)^1 \cdot (3C2) \times (1/2) = 3/16$
The 5th fly completed the quota (only 2 were captured from the 1st four flies)
$Q_2 = (1/2)^2 (1/2)^2 \cdot (4C2) \times (1/2) = 3/16$
Qquota = Probability that the spider made its quota of three flies before the 6th fly made the attempt to pass:
$Q_\text{quota} = Q_1 + Q_2 + Q_3 = 1/8 + 3/16 + 3/16$
$Q_\text{quota} = 1/2$ ← this is the probability that the spider will not attack the 6th fly
Pattack = Probability that the spider will attack the 6th fly
$P_\text{attack} = 1 - Q_\text{quota} = 1 - 1/2$
$P_\text{attack} = 1/2$
Pcapture = Probability that the 6th fly will get caught by the spider
$P_\text{capture} = P_\text{attack} \times P_\text{catch a fly} = 1/2 \times 1/2$
$P_\text{capture} = 1/4$
Hence,
$P_\text{survive} = 1 - 1/4$
$P_\text{survive} = 3/4$ ← answer
