Question

Can someone please help me with my homework, I'd really appreciate it.

1. 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%.)

a. 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.



b. Compute the probability that the first legitimate e-mail she finds is the fourth or fifth message she checks.



c. Compute the probability that the first legitimate e-mail she finds is among the first four messages she checks.



d. On average, how many messages should she expect to check before she finds a legitimate e-mail? (Round your answer to one decimal place.)

2. An actress has a probability of getting offered a job after a try-out of 0.08. She plans to keep trying out for new jobs until she gets offered. Assume outcomes of try-outs are independent.
Find the probability she will need to attend more than 7 try-outs.

Answer 1

Let x be the number of failures before first success ( number of spams before getting first legitimate )

p is the probability that message is legitimate = 0.10

x follow geometric distribution with p = 0.10, q = 0.90

P( x ) = q^x*p ; x = 0,1,2,3,...

a) P( fourth message is legitimate ) = P( x = 3 )  

=(0.90)³*0.1

= 0.0729

b) P( 4^th or 5^th legitimate message ) = P( x =3 ) + P( x = 4 )

= 0.0729 + (0.90)⁴*0.1 = 0.0729 + 0.0656

= 0.1385

c) P( legitimate among the first four messages ) = P(0) + P(1) + P(2) + P(3)

= 0.10 * [ (0.90)⁰+ (0.90)¹ +(0.90)² + (0.90)³ ]

= 0.3439

d) Expected legitimate mails = q / p = 0.9/0.1

Expected legitimate mails =  9

2) Let x be the number of failure before first success ( number of rejected jobs before getting first offer )

p = 0.08 , q = 0.92

x follows geometric distribution with p = 0.08 and q = 0.92

P( more than 7 try-outs. ) = P( x = 8 ) + P( x = 9 ) + P( x = 10) +...

= 1 - P( x ≤ 7)

= 1 - [ P(0) + P(1) + P(2) + P( 3) + P(4) + P(6) +P(7) ]

= 1 - 0.08 [ 0.92⁰ + 0.92¹ +0.92² +0.92³ +0.92⁴ +0.92⁵ +0.92⁶ + 0.92⁷ ]

= 0.5444