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.)
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 seventh message she checks.
Compute the probability that the first legitimate e-mail she finds is the seventh or eighth message she checks.
Compute the probability that the first legitimate e-mail she finds is among the first seven messages she checks.
On average, how many messages should she expect to check before she finds a legitimate e-mail?
REALLY CONFUSED. DONT KNOW WHERE TO START.
I BELIEVE THIS IS GEOMETRIC DISTRIBUTION.
You are correct that this problem can be solved using the geometric distribution. The geometric distribution models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials, where each trial has the same probability of success.
In this case, we have two possible outcomes for each trial: the e-mail is either legitimate (success) or spam (failure). The probability of success (P) is given as 0.10 (10% of the e-mails are legitimate) and the probability of failure (1 - P) is given as 0.90 (90% of the e-mails are spam).
Let's answer each part of the question step by step:
1. Probability that the first legitimate e-mail she finds is the seventh message she checks:
To find this probability, we need to calculate the probability of having six consecutive spam e-mails and then the seventh e-mail being legitimate. Since each trial is independent, we can multiply the probabilities together.
P(6 consecutive spam e-mails) = (0.90)^6
P(legitimate on seventh e-mail) = 0.10
Therefore, the probability is:
P = (0.90)^6 * 0.10
2. Probability that the first legitimate e-mail she finds is the seventh or eighth message she checks:
To find this probability, we need to calculate the probability of either having six consecutive spam e-mails and then the seventh e-mail being legitimate, or having seven consecutive spam e-mails and then the eighth e-mail being legitimate. Again, since each trial is independent, we can add the probabilities together.
P(6 consecutive spam e-mails and legitimate on seventh) = (0.90)^6 * 0.10
P(7 consecutive spam e-mails and legitimate on eighth) = (0.90)^7 * 0.10
Therefore, the probability is:
P = (0.90)^6 * 0.10 + (0.90)^7 * 0.10
3. Probability that the first legitimate e-mail she finds is among the first seven messages she checks:
To find this probability, we need to calculate the complement of the probability that all the first seven e-mails are spam.
P(all spam in first seven e-mails) = (0.90)^7
P(first legitimate e-mail among first seven) = 1 - (0.90)^7
4. Expected number of messages she should expect to check before finding a legitimate e-mail:
The expected number of trials until the first success in a geometric distribution is given by:
E(X) = 1 / P
Therefore, the expected number of messages she should expect to check is:
E(X) = 1 / 0.10 = 10
So, on average, she should expect to check 10 messages before finding a legitimate e-mail.
I hope this explanation helps clarify the steps needed to solve this problem using the geometric distribution.