Question

(A) Three marksmen fire simultaneously and independently at a target. What is the probability of the target being hit at least once, given that marksman one hits a target nine times out of ten, marksman two hits a target eight times out of ten while marksman three only hits a target one out of every two times. (B) Fifty teams compete in a student programming competition. It has been observed that 60% of the teams use the programming language C while the others use C++, and experience has shown that teams who program in C are twice as likely to win as those who use C++. Furthermore, ten teams who use C++ include a graduate student, while only four of those who use C include a graduate student. (a) What is the probability that the winning team programs in C? (b) What is the probability that the winning team programs in C and includes a graduate student? (c) What is the probability that the winning team includes a graduate student? (d) Given that the winning team includes a graduate student, what is the probability that team programmed in C? (C) A brand new light bulb is placed in a socket and the time it takes until it burns out is measured. Describe an appropriate sample space for this experiment. Use mathematical set notation to describe the following events: (a) A = thelight bulb lasts at least 100 hours. (b) B = thelight bulb lasts between 120 and 160 hours. (c) C = thelight bulb lasts less than 200 hours. (D) A university professor drives from his home in Cary to his university office in Raleigh each day. His car, which is rather old, fails to start one out of every eight times and he ends up taking his wife’s car. Furthermore, the rate of growth of Cary is so high that traffic problems are common. The professor finds that 70% of the time, traffic is so bad that he is forced to drive fast his preferred exit off the beltline, Western Boulevard, and take the next exit, Hillsborough street. What is the probability of seeing this professor driving to his office along Hillsborough street, in his wife’s car?

Answer 1

A) P(Marksman 1 hits a target) = 9/10 = 0.9

P(Marksman 2 hits a target) = 8/10 = 0.8

P(Marksman 3 hits a target) = 1/2 = 0.5

=> P(None of the markmen hit the target while firing simultaneously and independently)

= (1 - 0.9) * (1 - 0.8) * (1 - 0.5) = 0.01

=> P(At lease 1 markman hit the target while firing simultaneously and independently)

= 1 - 0.01 = 0.99

B) P(A team programs in C) = 0.6 ---- (1)

P(A team programs in C++) = 1 - 0.6 = 0.4 ---- (2)

=> No. of teams that program in C = 0.6 * 50 = 30, No. of teams that program in C++ = 0.4 * 50 = 20

P(A team programming in C wins) = 2*P(A team programming in C++ wins)

But, P(A team programming in C wins) + P(A team programming in C++ wins) = 1

=> P(A team programming in C wins) = 2/3, P(A team programming in C++ wins) = 1/3 ---- (3)

Also, given, P(A Team programming in C has a graduate student) = 4 / 30 = 0.133 ---- (4)

And, P(A Team programming in C++ has a graduate student) = 10 / 20 = 0.5 ---- (5)

a) P(Winning team programs in C) = P(A team wins / The team programs in C) * P(The team programs in C)

= 2/3 * 0.6 = 0.4 (using (3) )

b) P(Winning team programs in C and includes a graduate student)

= P(Winning team programs in C) * P(The team includes a graduate student / The team programs in C)

= 0.4 * 0.133 = 0.0532

c) P(Winning team includes a graduate student)

= P(Team programming in C wins) * P(Team programming in C has a graduate student)

+ P(Team programming in C++ wins) * P(Team programming in C++ has a graduate student)

= 2/3 * 0.133 + 1/3 * 0.5 (using 3, 4 and 5 above)

= 0.255

d) P(Winning team programmed in C / The team includes a graduate student)

= P(Winning team programs in C and includes a graduate student) / P(Winning team includes a graduate student)

= 00532 / 0.255 (using answers to b and c parts above)

= 0.209

C) An appropriate samples space for the life time of the bulb, denoted by T, is: {T: 0 < T < }

a) A = {T: T >= 100}

b) B = {T: 120 < T < 160}

c) C = {T: T <= 200}

D) P(Seeing the Prof. driving to his office along Hillsborough street in his wife's car)

= P(Professor taking his wife's car) * P(Prof. is forced to drive fast his preferred exit along Hillsborough street

= 1/8 * 0.7

= 0.0875