Homework 3: Probability Random Processess

Karthik V.Kalyani

September 2015

1 List of Questions

1. War game strategists make living by solving problems of the followingtype.There are six incoming ballistic missiles(BM’s) against which arefired twelve Anti-Missile Missiles(AMM’s).The AMM’s are fired so thattwo Amm’s are directed against each BM.The single shot killing Probabil-ity (SKKP) of an AMM is 0.8. The SSKP is simply the probability thatan AMM destroys a BM. Assume that an AMM’s dont interfere with eachother and that an AMM can, atmost destroy only BM against which itwas fired, compute the probability

• All BM’s are destroyed.

• Atleast one BM gets through to destroy the target.

• Exactly one BM gets through.

2. Assume in problem 1, that the target was destroyed by the BM’s what isthe Conditional Probability that only BM got through?

3. (Multinomial Probability Law) A computer chip manufacturer finds that,historically, for every 100 chips produced, 85 meet specifications, 10 needreworking, and 5 need to be discarded , Ten chips are sampled at randomfor inspection.

• What is the probability that all 10 meet the specifications?

• What is the probability that two are more need to be discarded?

• What is the probability that 8 meet specifications, 1 need reworking,and 1 will be discarded?

4. A smuggler , trying to pass himself off as a glass-bead importer, attemptsto smuugle diamonds by mixing diamond beads among glass beads inthe proportion of one diamond bead per 1000 beads. A harried customsinspector examines a sample of 100 beads. What is the probability thatthe smuggler will be caught, that is, that there will be atleast one diamondbead in the samples?

1

5. Assume that a faulty receiver produces audible clicks to the great annoy-ance of the listener. The average number of the clicks per second dependson the receiver temperature and is given by λ(τ) = 1−exp(−τ/10), whereτ is the time from turn-on. Device a formula for the probability of the0,1,2....clicks during the first 10 seconds of operation after the turn-on.Assume the Poisson Law.

6. The circuit shown in the figure below represents a telephone communi-cation link. Switches αi, i= 1, 2, 3...6 are open or closed and operateindependently. The probability that a switch is closed is p. Let Ai repre-sent the event that switch ’i’ is closed.

• Interms of the A′is write the event that there exists atleast one closed

path from 1 to 2 .

• Compute the probability of thers being atleast one closed path from

1 to 2 .

7. An automatic breathing apparatus (B) used in anasthesia fails with prob-ability pB . A failure means death to the patient unless a monitor system(M) detects the failure and alerts with probability pM . The failure of thesystems are independent events. Professor X, an M.D at H.M.S, arguesthat if pM>pB , installation of M is useless. Show that Prof. X needs totake a course on Probability Theory by computing the probability of apatient dying with and with out the monitor system in place. Take pM=0.1= 2pB .

2 Solutions:

2