Receiving Spam Emails

Solution & Explanation

To solve this problem, we need to determine the probability that at least 5 out of 100 received emails are spam. This is a classic binomial probability problem, where we have:

• Number of trials (n): 100 (the number of emails)
• Probability of success (p): 0.10 (probability of an email being spam)
• Number of successes (k): at least 5

Approach

1. Understand the Problem: We want to find the probability that at least 5 emails are spam. This can be expressed as $P(X \geq 5)$.

2. Use the Complement Rule: Instead of directly calculating $P(X \geq 5)$, it's simpler to calculate $P(X < 5)$ and then use the complement rule:

   $P(X \geq 5) = 1 - P(X < 5) = 1 - P(X \leq 4)$

3. Calculate $P(X \leq 4)$: This involves summing up the probabilities of receiving 0, 1, 2, 3, or 4 spam emails:

   $P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)$

4. Use the Binomial Probability Formula:

   The binomial probability formula is:

   $P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$

   Where:

   • $\binom{n}{k}$ is the number of combinations of $n$ items taken $k$ at a time, calculated as $\frac{n!}{k!(n-k)!}$.

5. Calculate Each Probability:

   • $P(X = 0)$:

     $P(X = 0) = \binom{100}{0} \cdot (0.1)^0 \cdot (0.9)^{100} \approx 0.00003$

   • $P(X = 1)$:

     $P(X = 1) = \binom{100}{1} \cdot (0.1)^1 \cdot (0.9)^{99} \approx 0.0003$

   • $P(X = 2)$:

     $P(X = 2) = \binom{100}{2} \cdot (0.1)^2 \cdot (0.9)^{98} \approx 0.0017$

   • $P(X = 3)$:

     $P(X = 3) = \binom{100}{3} \cdot (0.1)^3 \cdot (0.9)^{97} \approx 0.006$

   • $P(X = 4)$:

     $P(X = 4) = \binom{100}{4} \cdot (0.1)^4 \cdot (0.9)^{96} \approx 0.015$

6. Sum These Probabilities:

   $P(X \leq 4) = 0.00003 + 0.0003 + 0.0017 + 0.006 + 0.015 \approx 0.023$

7. Calculate the Complement:

   $P(X \geq 5) = 1 - P(X \leq 4) = 1 - 0.023 \approx 0.977$

Conclusion

The probability that at least 5 out of 100 received emails are spam is approximately 0.977 or 97.7%. This high probability suggests that receiving a significant number of spam emails is quite likely, given the parameters of the problem.