According to Internet security experts, approximately 90% of all e-mail messages
ID: 3207189 • Letter: A
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 fourth message she checks.
Compute the probability that the first legitimate e-mail she finds is the fourth or fifth message she checks.
Compute the probability that the first legitimate e-mail she finds is among the first four messages she checks.
On average, how many messages should she expect to check before she finds a legitimate e-mail? (Round your answer to one decimal place.)
Explanation / Answer
The probability that the first legitimate e-mail she finds is the fourth message she checks. = 0.93*0.1 = 0.0729
The probability that the first legitimate e-mail she finds is the fourth or fifth message she checks.= 0.93*0.1 + 0.94*0.1 = 0.0729 + 0.06561 = 0.13851
The probability that the first legitimate e-mail she finds is among the first four messages she checks. = 4C1p3q = 4*0.93*0.1 = 0.2916
On average, how many messages should she expect to check before she finds a legitimate e-mail = 1*0.1 + 2*0.9*0.1 + 3*0.92*0.1 + 4*0.93*0.1 + 5*0.94*0.1 + . ... .. = 0.1/(1-0.9)2 = 0.1/0.01 = 10