For safety reasons, four different alarm systems were installed in the vault containing the safety deposit boxes at a Beverly Hills bank. Each of the four systems detects theft with a probability of 0.99 independently of the others.The bank, obviously, is interested in the probability that when a theft occurs, at least one of the four systems will detect it.What is the probability that when a theft occurs, at least one of the four systems will detect it? (0.99)4 (0.01)4 1 − (0.99)4 1 − (0.01)4

For safety reasons, three different alarm systems were installed on the property of a famous movie star. Each of the three systems detects when a trespass occurs with a probability of 0.98 independently of the others.The movie star, obviously, is interested in the probability that when a trespass occurs, at least one of the three systems will detect it.What is the probability that when a trespass occurs, at least one of the systems will detect it?

For a particular bank's security system, the first alarm has a 95% chance of working while the second alarm has an 82% chance of working.   Use this information to answer questions #11 - 12.11)  Find the probability that BOTH alarms will work.        1 pointYour answer12)  Find the probability that AT LEAST one alarm will work.1 point

A homeowner has two smoke detector alarms installed, one in the dining room (adjacent to the kitchen) and one in an upstairs bedroom (above the kitchen). If cooking produces smoke in the kitchen, the probability of setting off the dining room alarm (D) is .95. The probability of setting off the bedroom alarm (B) is .40. The two alarms detect smoke independently of each other. If there is smoke in the kitchen, what is the probability that the smoke will be detected and will set off an alarm?Let's first understand what probability we need to find: The smoke from the fire is detected if it sets off the dining room alarm (D) or the bedroom alarm (B) or both, and therefore P(smoke is detected) = P(D or B), which is the probability we need to find. To that end, we are given a few pieces of information. Let's summarize them:To that end, we are given a few pieces of information. Let's summarize them:* P(D) = .95* P(B) = .4* Unlike the previous examples, in which P(A and B) was simply given, here we have a different piece of information: "The two alarms detect smoke independently of each other." In other words, instead of being given P(D and B), we are given the fact that D and B are independent.Explain why we have enough information to find P(D and B); then find P(D and B).

The probability of a security breach in a network is 5%. A security system detects breaches with 99% accuracy but has a 1% false positive rate. What is the probability that a detected breach is a false alarm?

70% of houses in a residential area have an alarm. It is found that 35% of the owners of thealarmed houses have a dog and 42% of the owners of the non-alarmed houses have a dog.a) Construct a tree diagram to show all possible events and probabilities.b) A house is selected at random. Calculate the probability that the house does nothave a dog.c) If a randomly selected house is one of the houses that does not have a dog, what isthe probability that the house has an alarm?