4.3 Problems on independence of events  (Page 4/5)

Three fair dice are rolled. What is the probability at least one will show a six?

A hobby shop finds that 35 percent of its customers buy an electronic game. If customers buy independently, what is the probability that at least one of the nextfive customers will buy an electronic game?

Under extreme noise conditions, the probability that a certain message will be transmitted correctly is 0.1. Successive messages are acted upon independently bythe noise. Suppose the message is transmitted ten times. What is the probability it is transmitted correctly at least once?

Suppose the class $\{A_i:1\le i\le n\}$ is independent, with $P\left(A_i\right)=p_i$ , $1\le i\le n$ . What is the probability that at least one of the events occurs? What is the probability that none occurs?

In one hundred random digits, 0 through 9, with each possible digit equally likely on each choice, what is the probility 8 or more are sevens?

Ten customers come into a store. If the probability is 0.15 that each customer will buy a television set, what is the probability the store will sell three or more?

Seven similar units are put into service at time $t=0$ . The units fail independently. The probability of failure of any unit in the first 400 hours is0.18. What is the probability that three or more units are still in operation at the end of 400 hours?

A computer system has ten similar modules. The circuit has redundancy which ensures the system operates if any eight or more of the units are operative.Units fail independently, and the probability is 0.93 that any unit will survive between maintenance periods. What is the probability of no system failure dueto these units?

Only thirty percent of the items from a production line meet stringent requirements for a special job. Units from the line are testedin succession. Under the usual assumptions for Bernoulli trials, what is the probability that three satisfactory units will be found in eight orfewer trials?

The probability is 0.02 that a virus will survive application of a certain vaccine. What is the probability that in a batch of 500 viruses, fifteen or more will survive treatment?

In a shipment of 20,000 items, 400 are defective. These are scattered randomly throughout the entire lot. Assume the probability of a defective is the same oneach choice. What is the probability that

1. Two or more will appear in a random sample of 35?
2. At most five will appear in a random sample of 50?

A device has probability p of operating successfully on any trial in a sequence. What probability p is necessary to ensure the probability of successes on all of the first four trials is $0.85$ ? With that value of p , what is the probability of four or more successes in five trials?

A survey form is sent to 100 persons. If they decide independently whether or not to reply, and each has probability 1/4 of replying, what is theprobability of k or more replies, where $k=15,20,25,30,35,40$ ?