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A cybersecurity system flags 30% of genuine login attempts as suspicious and correctly identifies 90% of hacking attempts as suspicious. On average, 0.5% of all login attempts are actual hacking attempts. If a login attempt is flagged as suspicious, what is the probability that it is actually a hacking attempt?

Answer :

To solve the problem of finding the probability that a login attempt is actually a hacking attempt, given that it is flagged as suspicious, we'll use Bayes' Theorem. Bayes' Theorem helps us update our probability estimates based on new evidence or information.

Let's break down the problem step-by-step:


  1. Define the Events:


    • Let [tex]H[/tex] be the event that a login attempt is a hacking attempt.

    • Let [tex]S[/tex] be the event that a login attempt is flagged as suspicious.



  2. Given Information:


    • The probability that a genuine login is flagged as suspicious: [tex]P(S | \neg H) = 0.30[/tex].

    • The probability that a hacking attempt is correctly flagged as suspicious: [tex]P(S | H) = 0.90[/tex].

    • The probability that any given login attempt is a hacking attempt: [tex]P(H) = 0.005[/tex].



  3. Calculate the Complementary Probabilities:


    • The probability that a login attempt is genuine (not a hacking attempt): [tex]P(\neg H) = 1 - P(H) = 0.995[/tex].



  4. Apply Bayes’ Theorem:

    Bayes' Theorem states:
    [tex]P(H | S) = \frac{P(S | H) \cdot P(H)}{P(S)}[/tex]

    To find [tex]P(S)[/tex], the total probability that a login attempt is flagged as suspicious, we use the law of total probability:
    [tex]P(S) = P(S | H) \cdot P(H) + P(S | \neg H) \cdot P(\neg H)[/tex]

    Substituting the values:
    [tex]P(S) = (0.90 \cdot 0.005) + (0.30 \cdot 0.995)[/tex]
    [tex]P(S) = 0.0045 + 0.2985 = 0.303[/tex]


  5. Calculate [tex]P(H | S)[/tex]:

    Now we substitute back into Bayes' Theorem:
    [tex]P(H | S) = \frac{0.90 \cdot 0.005}{0.303}[/tex]
    [tex]P(H | S) = \frac{0.0045}{0.303} \approx 0.01485[/tex]



So, the probability that a flagged login attempt is actually a hacking attempt is approximately [tex]0.01485[/tex] or 1.485%.

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