Question

According to Internet security experts, approximately 90% of all e-mail messages are spam (unsolicited commercial e-mail), while the remaining 10% are legitimate. A system administrator wishes to see if the same percentages hold true for the e-mail traffic on her servers. She randomly selects e-mail messages and checks to see whether or not each one is legitimate. (Unless otherwise specified, round all probabilities below to four decimal places; i.e. your answer should look like 0.1234, not 0.1234444 or 12.34%.) Assuming that 90% of the messages on these servers are also spam, compute the probability that the first legitimate e-mail she finds is the fifth message she checks:

Answer 1

The probability that the first legitimate e-mail is the fifth one checked, given a 90% spam rate, is 0.06561.

Step-by-step explanation:

To solve this problem, we need to understand the concept of geometric distribution. In this case, the event of interest is finding the first legitimate email, with a success probability of 10% (since 90% are spam). We want to find the probability that it occurs on the fifth check (meaning four failures followed by one success).

Here's how to calculate the probability:

1. Probability of a single check:

Success (finding a legitimate email): 10% or 0.1

Failure (finding spam): 90% or 0.9

2. Probability of four failures followed by one success:

The desired probability is the product of the probability of four failures followed by the probability of one success:

Probability of four failures: 0.9 × 0.9 × 0.9 × 0.9 = 0.6561

3. Probability of the first legitimate email being the fifth message:

This is essentially the product of the four failure probabilities and the success probability:

Probability = 0.6561 × 0.1 = 0.06561

Therefore, the probability that the first legitimate email the system administrator finds is the fifth message she checks is approximately 0.0656 or 6.56% rounded to four decimal places.

Answer 2

0.0656

Step-by-step explanation:

For each message, we have these following probabilities:

90% probability it is spam.

10% probability it is legitimate.

Compute the probability that the first legitimate e-mail she finds is the fifth message she checks:

The first four all spam, each with a 90% probability.

The fifth legitimate, with a 10% probability.

`P = (0.9)^(4) * 0.1 = 0.0656`