An automated cyber-security system scans 40 incoming packets for potential malware. For each packet, the probability of correctly identifying a malicious signature, independently of other packets, is 0.2. Let the random variable WWW denote the number of malicious packets correctly identified by the system.
Suggest a suitable distribution for WWW.
In a simplified reward model, the system gains 5 units of 'trust' for each malicious packet correctly identified and loses 2 units of 'trust' for each failure to identify a packet (assuming all 40 packets are actually malicious). Let GGG be the random variable representing the net gain in trust units.
Show that G=7W−80G = 7W - 80G=7W−80.
Find E(G)E(G)E(G) and Var(G)Var(G)Var(G).
Calculate the probability that the system achieves a net gain in trust of at least 5 units.
A high-traffic server processes a larger batch of 200 packets. For this server, the probability of identification is 0.15.
Using a suitable approximation, estimate the probability that the system correctly identifies at least 40 packets in this larger batch.
67 exam-style questions on OCR A Level Maths 2.4.4 Mean and variance of the binomial (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.