Data Visualization Homework

Probability of Detecting at Least one Intrusion Attempt

An intrusion detection system (IDS) monitors two independent network segments. Each segment can either detect an intrusion (D) or not detect any intrusion (N). Assuming that each segment has an equal chance of detecting or not detecting an intrusion (i.e., each has a 50% chance of D and 50% chance of N), what is the probability that at least one of the segments detects an intrusion?

Solving this,

P(at least one detection) = 1 – P(no detection in either)

Probability of detection (D) for each segment = 0.5

Probability of no detection (N) for each segment = 0.5

For no detection in either segment: P(N and N) = 0.5 × 0.5 = 0.25

Therefore, P(at least one detection) = 1 – 0.25 = 0.75

The probability that at least one segment detects an intrusion is 0.75 or 75%.

Probability of Multiple Malware Infections

A computer network experiences malware infections at an average rate of 3 per week. Assuming that malware infections follow a Poisson distribution, what is the probability that the network experiences more than 5 infections in a given week?

The probability of more than 5 infections means we need to subtract the probability of 0 to 5 infections from 1.

Using the Poisson formula with λ=3:

P(X > 5) = 1 – [e^(-3)(3^0/0! + 3^1/1! + 3^2/2! + 3^3/3! + 3^4/4! + 3^5/5!)]

= 1 – [0.0498 + 0.1494 + 0.2240 + 0.2240 + 0.1680 + 0.1008]

= 1 – 0.9160

= 0.0840

The probability of experiencing more than 5 malware infections in a week is approximately 0.084 or 8.4%.

Probability of Firewall Alert

A security analyst reviews alerts from two firewalls. Each firewall operates independently and has a 30% chance of generating an alert (A) indicating suspicious activity and a 70% chance of not generating any alert (N). What is the probability that at least one firewall generates an alert?

A = 0.3

N = 0.7

P(at least one alert) = 1 – P(N)

= 1 – (0.7 × 0.7)

= 1 – 0.49

= 0.51

The probability that at least one firewall generates an alert is 0.51 or 51%.

Conditional Probability of Specific Firewall Alert

Given that at least one firewall has generated an alert, what is the probability that it was the first firewall that generated the alert? Use the same probabilities as in Question 3 (each firewall has a 30% chance of generating an alert and operates independently).

P(first firewall alert | at least one alert) = P(first firewall alert AND any alert) / P(at least one alert)

From the previous question, we know P(at least one alert) = 0.51

P(first firewall alert AND any alert) = P(first alerts) = 0.3

Therefore:

0.3/0.51 = 0.588

The probability that the first firewall generated an alert, given that at least one alert occurred, is approximately 0.588 or 58.8%.

Probability of Multiple Transmission Errors

A data packet consists of 1,000 bits, and each bit has a 0.5% chance of being corrupted during transmission. Assuming that bit errors occur independently, what is the probability

For large n and small p, the binomial distribution can be approximated by the Poisson distribution with λ = np = 1000 × 0.005 = 5

P(X > 0) = 1 – P(X = 0)

= 1 – e^(-5)

= 1 – 0.0067

= 0.9933

The probability of experiencing at least one-bit error in the packet is approximately 0.9933 or 99.33%.

Probability of Selecting Vulnerable Servers

In a data centre with 40 servers, 15 are vulnerable to a certain exploit. If a security patch is deployed to 10 randomly selected servers, what is the probability that exactly 6 vulnerable servers are patched? Assume that servers are selected without replacement.

We need to find the probability of selecting exactly 6 vulnerable servers when randomly selecting 10 servers from a total of 40 servers, of which 15 are vulnerable.

The calculation uses the hypergeometric distribution formula:

[C(15,6) × C(25,4)] / C(40,10)

= (5005 × 12650) / 847660375

= 0.2205

The probability of patching 6 vulnerable servers is approximately 0.2205 or 22.05%.

Probability of Password Guessing on a Specific Attempt

A hacker is attempting to guess a user’s password, which is a 6-digit numeric code (from 000000 to 999999). Assuming each guess is independent and equally likely, what is the probability that the hacker guesses the correct password on their 20th attempt?

For each attempt, the probability of success is 1/1000000 (as there are 1000000 possible combinations)

The probability of failing 19 times and then succeeding on the 20th try:

(999999/1000000)^19 × (1/1000000)

= 0.00000099995

The probability of guessing the correct password on exactly the 20th attempt is approximately 0.00000099995 or about 0.0001%.

Probability of Successful Password Guessing Within Attempts

A malicious actor is trying to crack a 6-character lowercase alphabetical password (each character from ‘a’ to ‘z’). Assuming there are no restrictions on password selection and all passwords are equally likely, if they can make 1 million guesses, what is the probability that they successfully guess the password within those attempts?

The total possible passwords = 26^6 (26 choices for each of 6 positions)

= 308,915,776 combinations

P(success within 1 million attempts) = 1 – P(failure in all attempts)

= 1 – ((308,915,776 – 1,000,000)/308,915,776)^1,000,000

= 1 – (0.996764)^1,000,000

= 0.00323

The probability of successfully guessing the password within 1 million attempts is approximately 0.00323 or 0.323%.

Probability of a Specific Number of Phishing Emails Detected

An email security system detects phishing emails at an average rate of 5 per day. Assuming that phishing email arrivals follow a Poisson distribution, what is the probability that exactly 7 phishing emails are detected in a single day?

Uses the Poisson distribution to calculate the probability of exactly 7 events occurring when the average rate is 5 events per day.

Using the Poisson probability formula with λ=5:

P(X = 7) = (e^-5 × 5^7) / 7!

= (0.00674 × 78125) / 5040

= 0.1044

The probability of detecting exactly 7 phishing emails in a day is approximately 0.1044 or 10.44%.

Probability of Quick Cyber Attack

The time until the next cyber-attack on a system follows an Exponential distribution with an average time of 8 hours between attacks. What is the probability that the next attack will occur within the next 2 hours?

Using exponential distribution with λ = 1/8:

P(X ≤ 2) = 1 – e^(-2/8)

= 1 – e^(-0.25)

= 1 – 0.7788

= 0.2212

The probability that the next attack occurs within 2 hours is approximately 0.2212 or 22.12%.

Probability of Delayed Security Breach

Assuming the time until the 4th security breach follows a Gamma distribution with shape parameter α = 4 and rate parameter λ = 0.1 breaches per hour, what is the probability that the fourth breach occurs after more than 50 hours?

4th event occurring after 50 hours,

shape parameter α=4 and rate parameter λ=0.1.

Using the Gamma distribution formula:

P(X > 50) = 1 – P(X ≤ 50)

= 1 – [1/Γ(4) × ∫(0 to 50) (0.1)^4 × x^(4-1) × e^(-0.1x) dx]

= 1 –(0.1667 × 0.0001 × 47,1240)

= 1 – 0.7854

= 0.2146

The probability that the fourth breach occurs after 50 hours is approximately 0.2146 or 21.46%.

Applying Bayes’ Rule to Determine Attack Origin

A system can be compromised via three attack vectors: SQL injection (S), cross-site scripting (X), and buffer overflow (B). The prior probabilities of these attacks are P(S) = 0.5, P(X) = 0.3, and P(B) = 0.2. Given that an attack was successful and bypassed the intrusion detection system (event F), with probabilities P(F|S) = 0.1, P(F|X) = 0.4, and P(F|B) = 0.2, what is the probability that a successful attack was due to cross-site scripting?

Using Bayes’ Theorem to find P(X|F):

First, calculating P(F):

P(F) = P(F|S)×P(S) + P(F|X)×P(X) + P(F|B)×P(B)

= 0.1×0.5 + 0.4×0.3 + 0.2×0.2

= 0.05 + 0.12 + 0.04

= 0.21

Then, P(X|F) = P(F|X)×P(X)/P(F)

= (0.4×0.3)/0.21

= 0.12/0.21

= 0.5714

The probability that a successful attack was due to cross-site scripting is approximately 0.5714 or 57.14%.

Markov Chain Model of Security States

A system can be in one of three states: Secure (State 1), Under Attack (State 2), or Compromised (State 3). The transition probabilities per time step are as follows:

From Secure to Under Attack: 0.1
From Secure to Compromised: 0.02
From Secure to Secure: Remaining probability
From Under Attack to Compromised: 0.3
From Under Attack to Secure: 0.05
From Under Attack to Under Attack: Remaining probability
From Compromised to Compromised: 1 (the system remains compromised once it reaches this state)

This Markov chain problem requires constructing a transition matrix

Calculating the probability of moving from State 1 to State 3 in exactly 4 steps using matrix multiplication.

First, we construct the transition matrix P:

[0.88 0.10 0.02]

[0.05 0.65 0.30]

[0.00 0.00 1.00]

To find probability after 4 transitions, we compute P⁴:

P² (P × P):

[0.7864 0.1410 0.0726]

[0.0890 0.4775 0.4335]

[0.0000 0.0000 1.0000]

P³ (P² × P):

[0.7037 0.1688 0.1275]

[0.0905 0.3910 0.5185]

[0.0000 0.0000 1.0000]

P⁴ (P³ × P):

[0.6267 0.1965 0.1768]

[0.0920 0.3245 0.5835]

[0.0000 0.0000 1.0000]

The entry in row 1, column 3 of P⁴ (0.1768) represents the probability of moving from State 1 to State 3 in 4 transitions.

Therefore, the probability of moving from State 1 (Secure) to State 3 (Compromised) in exactly 4 transitions is 0.1768 or 17.68%.

Detecting Anomalous Network Traffic Using Relative Entropy

An analyst is comparing observed network traffic patterns to known profiles. The traffic consists of packet sizes classified into small, medium, and large, with observed frequencies:

Observed Traffic: Small = 0.25, Medium = 0.50, Large = 0.25
Normal Traffic Profile: Small = 0.40, Medium = 0.40, Large = 0.20
Anomalous Traffic Profile: Small = 0.20, Medium = 0.30, Large = 0.50

Using Relative Entropy (Kullback-Leibler Divergence), determine whether the observed traffic is closer to the normal or anomalous traffic patterns.

KL Divergence from Observed to Normal:

D(P||Q1) = 0.25 × log(0.25/0.40) + 0.50 × log(0.50/0.40) + 0.25 × log(0.25/0.20)

= -0.4055 + 0.2231 + 0.2231

= 0.0407

KL Divergence from Observed to Anomalous:

D(P||Q2) = 0.25 × log(0.25/0.20) + 0.50 × log(0.50/0.30) + 0.25 × log(0.25/0.50)

= 0.2231 + 0.5108 – 0.6931

= 0.0408

Since the KL divergence to the normal profile (0.0407) is slightly less than to the anomalous profile (0.0408), the observed traffic is marginally closer to the normal traffic pattern.

Probability of Intrusion Detection System Failing

An intrusion detection system (IDS) has a 2% chance of failing to detect an intrusion attempt during any given attack. If 150 attacks occur, what is the probability that the IDS fails to detect at least one intrusion?

P(at least one failure) = 1 – P(no failures)

= 1 – (0.98)^150

= 1 – 0.0470

= 0.9530

The probability that the IDS fails to detect at least one intrusion out of 150 attacks is approximately 0.9530 or 95.30%.

Estimating Time Until Next Zero-Day Exploit

The time between zero-day exploits being discovered follows an Exponential distribution with an average of one exploit every 60 days. What is the probability that the next zero-day exploit is discovered within the next 30 days?

P(X ≤ 30) = 1 – e^(-30/60)

= 1 – e^(-0.5)

= 1 – 0.6065

= 0.3935

The probability of discovering a zero-day exploit within 30 days is 0.3935 or 39.35%.

Probability of System Crash Due to Software Bugs

A critical software system contains 5 latent bugs. Each bug independently causes a system crash with a probability of 0.1 during a day of operation. If multiple bugs cause crashes on the same day, each crash is a separate event (i.e., the system can experience multiple crashes in one day). What is the probability that the system experiences exactly two crashes in one day?

P(X = 2) = C(5,2) × (0.1)^2 × (0.9)^3

= 10 × 0.01 × 0.729

= 0.0729

The probability of exactly two crashes in one day is 0.0729 or 7.29%.

Applying Hypergeometric Distribution to Security Audits

A company has 30 applications, of which 12 contain critical vulnerabilities. If an auditor randomly selects 6 applications for a security review, what is the probability that at least 4 of them have critical vulnerabilities?

P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6)

= [C(12,4)×C(18,2) + C(12,5)×C(18,1) + C(12,6)×C(18,0)]/C(30,6)

= 0.0540

The probability of finding at least 4 vulnerable applications is 0.0540 or 5.40%.

Using Bayes’ Rule for Malware Detection

An antivirus software correctly identifies malware with a true positive rate of 99% (i.e., P(Positive Test | Malware) = 0.99) and has a false positive rate of 2% (i.e., P(Positive Test | No Malware) = 0.02). If 1% of all files scanned are malware (i.e., P(Malware) = 0.01), what is the probability that a file identified as malware is indeed malicious?

P(M|P) = P(P|M)×P(M)/[P(P|M)×P(M) + P(P|NM)×P(NM)]

= 0.99×0.01/[0.99×0.01 + 0.02×0.99]

= 0.0099/0.0297

= 0.3333

The probability that a file flagged as malware is actually malicious is 0.3333 or 33.33%.

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Question

Probability of Detecting at Least One Intrusion Attempt

Probability of Multiple Malware Infections

Probability of Firewall Alert

A security analyst reviews alerts from two firewalls. Each firewall operates independently and has a 30% chance of generating an alert (A) indicating suspicious activity, and a 70% chance of not generating any alert (N). What is the probability that at least one firewall generates an alert?

Conditional Probability of Specific Firewall Alert

Probability of Multiple Transmission Errors

A data packet consists of 1,000 bits, and each bit has a 0.5% chance of being corrupted during transmission. Assuming that bit errors occur independently, what is the probability that more than 5 bits are corrupted in the packet?

Probability of Selecting Vulnerable Servers

In a data center with 40 servers, 15 are vulnerable to a certain exploit. If a security patch is deployed to 10 randomly selected servers, what is the probability that exactly 6 vulnerable servers are patched? Assume that servers are selected without replacement.

Probability of Password Guessing on a Specific Attempt

Probability of Successful Password Guessing Within Attempts

Probability of a Specific Number of Phishing Emails Detected

Probability of Quick Cyber Attack

The time until the next cyber-attack on a system follows an Exponential distribution with an average time of 8 hours between attacks. What is the probability that the next attack occurs within the next 2 hours?

Probability of Delayed Security Breach

Applying Bayes’ Rule to Determine Attack Origin

Markov Chain Model of Security States

A system can be in one of three states: Secure (State 1), Under Attack (State 2), or Compromised (State 3). The transition probabilities per time step are as follows:

From Secure to Under Attack: 0.1
From Secure to Compromised: 0.02
From Secure to Secure: Remaining probability
From Under Attack to Compromised: 0.3
From Under Attack to Secure: 0.05
From Under Attack to Under Attack: Remaining probability
From Compromised to Compromised: 1 (the system remains compromised once it reaches this state)

Question:

Construct the transition matrix for this Markov chain. What is the probability that the system moves from Secure (State 1) to Compromised (State 3) in exactly 4 transitions?

Data Visualization Homework

Detecting Anomalous Network Traffic Using Relative Entropy

An analyst is comparing observed network traffic patterns to known profiles. The traffic consists of packet sizes classified into small, medium, and large, with observed frequencies:

Observed Traffic: Small = 0.25, Medium = 0.50, Large = 0.25
Normal Traffic Profile: Small = 0.40, Medium = 0.40, Large = 0.20
Anomalous Traffic Profile: Small = 0.20, Medium = 0.30, Large = 0.50

Using Relative Entropy (Kullback-Leibler Divergence), determine whether the observed traffic is closer to the normal or anomalous traffic patterns.

Probability of Intrusion Detection System Failing

Estimating Time Until Next Zero-Day Exploit

Probability of System Crash Due to Software Bugs

Applying Hypergeometric Distribution to Security Audits

Using Bayes’ Rule for Malware Detection

An antivirus software correctly identifies malware with a true positive rate of 99% (i.e., P(Positive Test | Malware) = 0.99) and has a false positive rate of 2% (i.e., P(Positive Test | No Malware) = 0.02). If 1% of all files scanned are actually malware (i.e., P(Malware) = 0.01), what is the probability that a file identified as malware is indeed malicious?

Data Visualization Homework

Data Visualization Homework

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